I tend to expect it to close. I shouldn't have to open task manager and force close an element of Galaxy that wants to overstay it's welcome. This whole thing with all these proprietary launchers feeling entitled to stick around and do god knows what in the background after the main program has been closed is not ok. Galaxy, Origin and Epic all do so. Steam is literally the only one that closes when I close it. Be transparent about your intentions with your software.

Nothing should be left running in the background, if you either click the cogwheel or right-click the systray icon and choose Exit. The X "Close" button in the top right will just minimize the window to the systray though.

If GOG Galaxy components hang on exit, you can click the cogwheel and choose either Report a Bug or Get Support.