Just because it shows the Force Stop button doesn't mean it's running, merely that at least one of its components is loaded. That can be just about anything. I have apps I know for a fact cannot run in the background that shows the force stop button.
Mainly, it boils down to battery management and the Android architecture. Android apps are very modular, so the Java class for handling push notifications might be loaded but none of its screens or other services would be loaded and it uses negligible amounts of memory. It's way more battery efficient than reloading it from storage, and if the system needs memory it'll clear some caches.