It's entirely possible inside an app to implement a mute feature.
You just have a flag that prevents sound from being played by the app when it otherwise would.
The fact that the app makes sound at all is something that had to be built, so somewhere in the source they are initializing a media player. They just need the ability to fully stop the media player in the app as opposed to just decreasing its volume.
To oversimplify a little, assuming this is written in Java (Likely given that it's on android, though it could be kotlin)
Something like this would do:
onMuteClicked(boolean mutedState){
if(mutedState == true){
this.mutedState = true;
this.mediaPlayer.stop();
} else{
this.mutedState = false;
this.startSound();
}
}
This is oversimplified, mind you. I don't actually know how the application is structured, but essentially you just need to detect when the state of the mute toggle is changed. If it is changed to true, stop the MediaPlayer objects you have running, and prevent them from starting while muted.
If changed to false, do whatever is normally done to initialize sound.
In whatever the startSound() function would be, you wrap the existing code in a block that only executes if this.mutedState is false.
Totally achievable.