So Firefox on Android does support PWA and notifications.
Not sure what device you use specifically
@sbj but I just witnessed some fairly odd and inconsistent behaviour between testing on a physical Samsung S20 and a Google Pixel 7 Pro via Browserstack (a real device, but remotely accessed).
On the latter, I installed the app, I accepted notifications and observed that no notifications were being received. I then went back to Firefox itself (not the PWA) and there was a message on screen again asking me to allow notifications (something I had already done).
After that, notifications still weren't readily apparent, but it turns out they were actually being received but they were "Silent" so they were kinda hidden at the bottom of the list of notifications.
So it's likely they're just not "allowed" or otherwise hidden.
As for app badge updates, that isn't supported by Firefox on Android:
caniuse.com
For all intents and purposes though, even if Firefox is your preferred browser, you might want to consider installing the PWA through Chrome. Still use Firefox for normal browsing but the PWA will use Chrome behind the scenes.