On my iPhone Anki is showing the wrong badge card count on the Anki icon. And is missing from notifications list in iPhone settings. Deleting and reinstalling, and a re synch still does not show in the notification section of settings.
On my iPad the badges work correctly and show as an option notifications in settings.
My Anki badge count (v 20083.1 on iOS) won’t update after I study unless…
When done studying, instead of closing the app, I switch to the home screen first.
Two seconds later the badges will update & Anki can be closed.
The badge is updated after the app is placed in the background. If by “closing the app” you mean you are forcing it to shut down, you should not be doing that unless it gets stuck.