When trying to reproduce the issue in this question, I noticed that some apps lack the "Allow app to run in the background" and the "... even when Battery Saver is on" options in Battery Saver. At first I thought those apps weren't even meant to run in the background, but then I noticed one of the apps lacking those options is Facebook Messenger, which clearly does run in the background (since I get notifications about new messages).
So does that mean I'm unable to prevent Facebook Messenger from running in the background? Or does it somehow piggyback on the main Facebook app (which does have the option and which I've allowed to run in the background)? What exactly determines whether or not the option(s) in question show up?