I only know pausing games is funky because the highest my playstation fans ever go is pausing some games. Quite weird pausing is not just a feature of the game engine or runtime, especially as the menu and settings system seem to be totally separate in most cases anyways.
It is a feature of most game engines. Unreal has a standard SetGamePaused function, for example.
But like most things in game development there is no solution that fits every use case.