Firefox supports a quite faithful port of the `chrome` object the Chrome extension APIs are namespaced under. The four extensions I've written in the past (generally internal-to-co) were all seamlessly supportable in both browsers once Firefox moved away from their old extension interface. There are some differences but Firefox keeps their additions/improvements they make contained to the `browser` namespace, which seem to be spec-driven[1] so maybe those changes will end up in Chrome some day too.
TL;DR: don't let worrying about compatibility prevent you from just trying to install and run the same code in FF to see what happens.
Also, is the source code for this publicly available? I was going to just check this myself, but I don't immediately see a GitHub link in this thread or at the linked page from FF for Android.
I tried it on Firefox and it works fine! I just needed to change the value of <secret_key> to a key I created on their website (or else it fails silently, it would be cool to have some console message I think).
Probably a little late as you've already shipped Chrome, but webextension-toolbox is a pretty neat little CLI tool to help write cross-platform extensions:
Mostly. There are some APIs that are supported by one platform and not the other, but the majority is shared. So there's a large chance this extension should just work mostly out of the box in Firefox. However, it still needs to be repackaged and released for each platform separately.