I do not believe it does quit it properly on OS X. Even after a fresh reboot, Spotify appeared to be using about 40k/sec outbound bandwith. After I removed it, idle bandwidth dropped to zero.
I also noticed something strange when removing it with CleanMyMac (similar to AppZapper). There's a separate dock icon file for Spotify, and removing it restarted Dock.app. I think that might be to cover up Spotify running in the background somehow.