In general, notifications on Android don’t go through apps, that would require apps to run all the time and consume too much battery. They instead are going through a notification provider (google in the default case). This is why your notifications arrive, but you have no other connections to your HA.
Edit for clarification: HA uploads it to google servers (which of course doesn’t need port forwarding) and your phone then polls them from google.
It’s my turn to do the obligatory mention of SourceHut :)
It is in alpha, but it is really promising. It is going all-in on email based git workflows (which was the original way of doing it before the github-style PR based workflow). I love the style and it’s minimalism - but don’t let that fool you, it has many features that you might not see at first glance. Imagine if cgit or gitweb was extended into a software forge with built-in support for email patches, mailing lists, issue tracking and CI.
If you are the type of person who attracts garbage issue tickets and often has to reject low-effort PRs on your projects, it forces a really good minimum entrance bar. Of course this comes at the cost of visibility of your projects, less networking effect, so I would suggest to not use it if you want easy visibility and 3rd party contribution on your projects.
It might be not this simple in case of notification providers, the notifications are somehow sent in the name of the app I guess. And don’t forget: in general, this principle is true for most of your apps that send notifications. I.e. if you are getting Signal or Facebook notifications, it uses the same principle.
Btw, google is not the only notification service for android. Check up on Unified Push, there are many alternatives to do this, and there are some apps that support these alternatives in their non-play-store builds.