The app has to make a request from your Lemmy instance to get notifications. For "real time" notifications it would have to be requesting updates constantly, which wouldn't be appreciated by instance owners and could maybe even get you blocked.
Boost For Lemmy
Community of the Android app Boost for Lemmy
You've got issues if you need rapid notifications while using the Fediverse/Lemmy... etc.
Take a chill pill.
I just asked for that as an option. Sometimes some people need timely notification for a particular post so they can set it.
Lemmy would need to push notifications to you from their servers for that. Periodically checking your inbox for messages is the best any app can do on their end. It was the same with Reddit, where they would only push notifications to the official app, and anything else would be limited to checking on its own.
Increasing the frequency of checks would also have an impact on your device's battery life and increase traffic for your instance.
Google and Apple both provide push notification services to avoid the polling problem.
But that would require each and every instance to register with each and every app for it to work, which is definitely not going to happen.