The option ‘App Icon Shows Due Count’ is on
(within the Anki app under Preferences > Review > Notifications)
But when new cards/reviews are due, the Anki icon doesn’t update itself - it only updates itself after I reopen the app.
(using iPhone 5s, on iOS 12.4.8)
When you exit the app, it calculates the counts for tomorrow. The counts should update early the next day provided you don’t turn your phone off, but will not update on subsequent days if you don’t return to the app, and will not update if you have learning cards due later on in the day.