It seems very likely that Firefox would be making use of GCM/FCM - I think most major apps do - so this could be it.
Incidentally, this same topics is one of the main controversies around the development of the Signal chat client - it uses GCM, which a good few prospective users object to on similar grounds.
Unfortunately, unless you've rooted your phone, GCM/FCM is the only battery-efficient way to provide push to Android phones.
But, and this is the interesting part, the userspace library of FCM can be easily reversed and cloned (the information required for that is here https://gist.github.com/justjanne/153bd4886f92be0ee57fb40842...), and the system-side part of FCM has been cloned in an open way (see the µG project).
Note: I should possibly have been clearer above by saying "is the only battery-efficient way to provide push to Android phones without root"