There could be many possible reasons for that but basically, whenever battery saver is on, it doesn't mean it is active, battery saver will be activated only when battery is low and it in that time that the app that is allowed to run in background will run.
To be more specific, almost every app is disabled in battery saver, you will have to specifically enable for it to run in background.
It guess it could be some misunderstanding with the settings, because I use it all the time that way and never had an issue.
mmm i think i've experienced this too but only in certain conditions.
with battery saver on and working i keep receiving Whatsapp messages only when i open the app and after using it lock the phone without going to my home screen first. If i press the windows key or the back arrow no notifications pop up when the phone is locked.