I don't know exactly what the option is called but since Android 8 there is at least a toggle there per app. Later versions have lots more settings.
Unused apps also indeed go in different App Standby buckets, which while it shouldn't affect FCM notifications, it does on some older android versions (https://developer.android.com/topic/performance/power/power-...)
If you're not using FCM, well you're limited to having to check for notifications yourself, either within doze windows, or by registering a foreground service that keeps your app alive and checking.