Its even worse when you force Firefox to use wayland its icon doesn't even show.
Edit: Oh since everyone now is confused; I only have the flatpak version of Firefox installed yet it doesn't use the pinned icon and doesn't even use the firefox icon under wayland at all.
In that particular screenshot I believe you’re right: the one on the left is Firefox ESR while the icon on the right is whatever flatpak version available.
But I know what OP is referring to as it is a open bug currently, the DE don’t doesn’t recognize the launched instance as the pinned program due to the way Flatpak launched apps. Not an issue with Firefox in particular
I actually took the screenshot myself and yes it is a bug* specifically with Flatpak.
I am having the same thing at the moment with the Firefox snap package under Ubuntu. Except as well as this, when it updates it seems to take out everything else pinned to the task bar with it. Maybe it's not Firefox doing that, but since I stopped pinning FF it has stopped happening.