Sort of. If you’re receiving a notification from a remote server on iOS or standard android, they go through Apple or googles servers. That said, some apps rather than sending your device the actual notification (where this vulnerability comes from) will instead send a type of invisible notification that basically tells the app to check for a new message or whatever and then will display a local notification so the actual message stays on device and inside of the hosting services servers (like a self host.)
That said, some apps rather than sending your device the actual notification
Pretty sure that is actually the recommendation from apple/google, as it reduces bandwidth for their notification servers.
I think the message payload is severely limited.
Like, pre-ios8 the limit was 256 bytes. Now it's 2kb.
Notify (hope I remmeber the name right) has an option for both push notifications (with the usage of Google services) and polling based notifications (fully self-hosted)