The way is to create custom Service https://developer.android.com/reference/android/app/Service.html , which is holding connection via socket or polling some remote server with intervals.
But, I suggest that Firebase service is more advanced and optimized in battery consumption and, particularly, in utilizing the sleep mode. It’s not trivial to make a service which will have fast response and CPU consumption. I think that a custom service development will lead to native (C++) coding of networking routines libs.
Also, take into account that Firebase provides own Authentication service with ability to seamless convert your anonymous user to registered. There is a lot of pros of Firebase services. And cons are: your users have to install Google Play Services, if your app will grow – you will pay to Firebase.
Look here Android push notification without Google service