Update:

Installed Instagram, Whatsapp (linked divice) and Telegram and tested the notifications and all worked fine. Can't test iMessage since I don't use an iCloud account in the phone.

I have this iPhone basically as an iPod with Youtube. YT Music and Spotify so can't do a proper test since is not my daily phone.

I think may be some problem with the Apple servers since you a seeing random notifications but I just can't discard that Apple really killed the SE1.

Thanks for checking.

However mine also work fine at first, but gradually worsens over a few days.
Erase and reinstall starts the process again.

All was fine on both phones until 2 weeks ago.