The difference is what code runs on your device. If proprietary libraries are included, F-Droid won't build it, and it's not allowed in their repository. There's a lot to say about whether a FOSS app that relies on proprietary network services is truly "free", there's no arguing that an app with proprietary code blobs is "free".

Take for example an app like NewPipe. The application itself doesn't include proprietary code, but it contacts YouTube, a proprietary Google service. With the app itself being open source, you can tell exactly what it is doing on your device, and what information is sent over the network. Comparing that to something like Signal, which includes proprietary Google libraries, you'd have to decompile and reverse engineer it to try and figure out what it's doing.

If you have a FOSS library that interacts with Google Play Services or microG to enable FCM, it would (probably) be allowed on F-Droid. (I'm not on their team, I can't make a definitive statement about this).

Ah, you are right, I installed Element from the Google store (using Aurora).

But Mastodon is from F-Droid and it uses Google notification service, I just checked.