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create my own push notification service without fcm , pusher ,

I want to create a notification system provider that is not based on Google services or similar. In fact, I want to get information about its overall architecture and needed Android sdk functionality.
The most weird point for me is to understand how to send a notification to an Android device.
I mean, how can I identify the Android device on which my application is installed from millions and millions of other Android devices on the Internet?
And how do I send information to him?
Should I use sockets for this or similar stuff?

simple answer: YOU CAN NOT
before everything else i should correct your question, pusher and FCM are not in the same group at all! you can build somthing like pusher or oneSignal or etc but you can not build something like FCM/APNS
you should understand three simple yet important sentences below:
when you want to pull anything from place_1(e.g. api) to place_2(e.g. browser_client) you most have an identifier of the place_1_resource (which commonly is the uniform-resource-locator of api)
when you want to push anything from place_1(e.g. notification_central_server) to place_2(cellphone_client) you most have an identifier of the place_2_resource
you must know the differences between a real server push with server-push-like technologies like long-pulling or ... and you should be aware that what is intended in this concept is a real server push not any kind of pulling with a push jacket!
if you don't have any identifier for a cellphone which you want to send it a notification, your server dont know where to send that notification so we need a resource_identifier_like for cellphones which is actually a device_token_like and you have just one approach to get this device_token_like and that is the FCM/APNS
FCM is like a dns server containing all identifiers of every android device that google supports (almost every android device) and APNS is just the same but for apple devices
note1: even if your app can obtain it's corresponding device device_token_like it can not be used for push notification if its not registered on FCM/APNS
so when you get that device_token_like identifier of your desired clinet_device now you can use different approaches for sending sth to that clinet_device. there are several approaches like SSE, Webpush, HTTP_server_push, Pushlet and etc but none of these approaches supported by mother_companies of these devices, the only approach that is completely supported and standard is the same approach that FCM/APNS official websites suggests
for example an iranian Incorporation named najva uses webpush to send notifications because of USA sanctions but webPush method works good on browsers and android devices but they didn't even apear on an apple devices
finally i should say that i admire your curiosity to less using anything from a benefit_based Inc. like FCM/APNS in your developing but i strongly recommend these articles and books for you cause i think you didn't learn enough:
wikipedia of push technology
story of some guy who tries to make his own push notification service
Push Technology A Complete Guide - 2020 Edition
Data Push Apps with HTML5 SSE

Short, direct answer
You can't (At least till you create your own ROM)
TL;DR, Reason why?
Before you build your own push notification server, you first need to know how it works internally in android.
Whenever you/your server sends a push notification message to the android client, the SDK processes it and shows you the notification. But when your app is not running (or being killed), your app cannot respond to it since it was not running. In such a case, your notification message is sent to a system service which is known as Google play service. For this even to work, you will first need to bind your app with Google play service and that is what FCM does. FCM SDK registers your app to the operating system service on the first initialization. That FCM service is opened to a port which listened to the incoming message from the server and when it receives the message, it publishes a notification on behalf of your app with a PendingIntent containing the data. Then the PendingIntent is delivered to your app when the user clicks it and then finally your app process the data (or the push message)
So basically, for your server to communicate with the client, It first needs to communicate with the FCM service and for that, FCM gives you a token which identifies the application to register with the internal Google play service.
Simplified furthermore, the workflow is as follows:-
Server send push message ---> FCM ---> Google play service,
If your app is running, it is directly handled by the client SDK So, Google play service --> Your app
If not, then it is delivered by the service itself using PendingIntent So, Google play service --> PendingIntent --->| Publish notification
Totally impossible, Workaround?
There is nothing like impossible because an absolute impossibility doesn't exist. Saying impossible generally means near to impossible. (This is similar to math where also we say tends to infinity because no one has achieved it yet).
To make it work, you need to somehow bind your app to the Google play service and you can't because Google hasn't exposed any direct API to do that. The only possible way is using the FCM ;-) (Bad luck again). So the only possible way is to build your own custom ROM with a custom push service that acts as a client for your Push server and a Server for your Push client (which is your app).
Since the above option tends to impossibility, you have to choose a workaround.
The best among the worst workarounds are:-
To make a malicious SDK.
Malicious because it needs to keep the app running in the background with a service that is connected to a WebSocket endpoint of your server. (Harder in new android versions).
Make use of a database where your push notification is saved and your app checks it periodically using AlarmManager.
Hope you have got the point.

Trying to setup device-to-device notifications via FCM; any recommendation when it comes to FCM and choosing XMPP server?

I read the documentation for FCM in Firebase website, looked into answers here for related questions and googled. All good, but as a newbie I am not sure what's best approach when it comes in choosing the XMPP server.
Is there anything that would limit one of the servers in the list blow when it comes to connecting to FCM? I have a list of servers here.
Should I choose one of the servers in this list for FCM if I need device to device notifications or is there any other choice. I need it for when a new message arrives in a chat app or similar cases when a ChildAdded or modified and the app is backgrounded or closed?
I cannot find any info on the XMPP servers and if Firebase has any recommendation. I would need all three iOS, Android, and Web app support.

I am using Google Cloud Messaging for push notifications in my app and i have 2 basic doubts about it

Is GCM the only way for push notifications in android? And is it advisable over other methods (if there are any)?
So let's say my app from a device registers on GCM. Is it the app itself which is assosciated with the device that registers itself, or the user and his email id that register? If my user switches devices in future, will the same 'registration token' work for him ? Or in a similar scenario if a user with another g-mail id registers to my app from the same device, do i push notifications to the same device with 2 different registration tokens?
Extending to what i asked in 2, suppose a single user has his g-mail id working on my app on different devices. Does GCM push the notification to all those devices, or only one of them? And if one, then how does it prioritize?

No, there are other push notification services in Android, like Parse. However, it is strongly recommended to use the default push notification service that is present in any Android device that has Google Play Store, because that reduces battery usage. If you are pushing your app somewhere else though, like Amazon Store, keep in mind that Kindle Fire does not comply with this and doesn't have Google Play Services -- at least as far as I know.
The registration token is very volatile and is not bound at all to an email. Actually, sometimes the same device may get a new GCM ID out of the blue. There are many tutorials on GCM's best practices, but Google's is the best way to start. What you should do to prevent problems is store the GCM id of your device in your database, so you can always know which ID is currently associated to your device. GCM may give you new IDs for no reason, so always parse its response properly and store the new ID it may provide, linking it to the device.
GCM doesn't really care which Google account is your app using. That means you can register as many devices as you want with the same account. You have to manage to which devices you push notifications, because the ID is managed by device.
Be careful: there's been some changes with the API lately, so a lot of unnoficial tutorials you find on the internet may point to deprecated methods.

There are other ways of sending notifications, Tokodu and RabbitMQ for example. I wont go over if X is better than Y, since it's not allowed on StackOverflow.
A unique device token is generated for the device, how you use this device token is totally up to you. You can make it unique for the user by, for example, implementing a login system, so that the device token is created and stored in a database (together with the user information). Let's say if two different users uses your application on a shared device, they can both receive notifications on the same device if you allow them to. It depends on your own implementation.
This again depends on how you chose to implement the GCM features on your client and server side. For example, you can choose to store device tokens for each and every device the user has installed the app on, or you can store the latest device token so that notifications are only sent to one of the user's devices.
I hope this helps clear some things up.

Besides good answers above, if you need a push notification solution for a working environment without Internet access, I suggest SignalR of Microsoft, which has already supported Android and Java (you can find out more at GitHub).
Hope this helps little!

send and receive notifications in android

I am developing an app in android in Eclipse IDE. The app modality requires that any user can challenge his/her friend to do a certain activity. Whenever a user does so, the friend will immediately get a notification in his/her phone. Even more, the user who challenged the friend, will also receive a notification when the friend accepts the challenge. Is there any particular way I can provide this functionality? I know how to generate notifications. But the server side interaction is quite confusing. Should I check the server database through the app at frequent intervals and check for such activities in database? I think the procedure will hamper the performance of the app. I would be very grateful if anyone can suggest a good way.

But the server side interaction is quite confusing.
As you're pointing out, it's the server part not the Android one what's confusing here. If you don't have any constraints on the server part, I'd recomend to use Socket.io http://socket.io/ They have a tutorial for connecting socket.io with Android: http://socket.io/blog/native-socket-io-and-android/

It seems that you don't really need an interaction between the differents users but only a way to notify them. I always used Google Cloud Messaging to do so. It enables you to receive push notifications. As you said, checking the server periodically is battery/data consuming and not the best way to do it.
This link might help to set up GCM : http://www.androidhive.info/2012/10/android-push-notifications-using-google-cloud-messaging-gcm-php-and-mysql/.

Android - Push Notifications

I am currently building a messaging application that allows users to send and receive messages on their Android mobile phones over an internet connection to each other. I have decided that I do not want to use polling because it means that a user may not receive another user's messages as instant as possible. I have my own server available for use.
I am currently tied between using Google's Cloud Messaging for Android platform in order to send the notifications from the server to the Android device. The other option is to keep a live TCP connection between my server and the Android device via a service, and send 'keep alive' messages every 5 minutes for example.
From your best opinion, what is the best way to do this - or is there a better way? I don't want to use third parties apart from Google to do this. There are similar answers available, but they don't address this specifically.
Alex

using Google's Cloud Messaging for Android platform in order to send the notifications from the server to the Android device.
This is not a realtime notification either; the notifications may be delayed longer than you experienced with polling. Also GCM is meant for broadcasting messages to a number of users, not for targeting a message to one specific user.
The other option is to keep a live TCP connection between my server and the Android device via a service
I don't know how many users you are expecting, but this may not scale. You are limited in the number of TCP connections to one server.
No need to re-invent the wheel here, use an existing implementation such as XMPP.

Take a look at this:
https://pusher.com/docs/client_libraries
https://github.com/pusher/pusher-test-android
It may be what you're looking for