agree with Attila, It depends on your needs, you need to answer these questions to find the best suitable solution. Also, For this kind of apps, I would recommend using Node Js and Socket io. I've worked on a very similar App and the best solution to handle our own requests is to build our service handler. Our stack was PHP Laravel, MySql, Node Js, Socket io, and Redis For some caching. using the pre-ready services most of the time has many limitations and very costly.

I’ve used both Pusher and Firebase for a real-time app. My recommendation is to use Firebase because it has better support and has additional features you might want in the future like offline caching, ML, Analytics, etc...