Shadow root (DOM) has been available in Chrome for a lot longer than Firefox (launched, possibly with bugs affecting extensions, at the end of 2018).
If you’re using the latest version of Firefox and Chrome the shadow roots might not be having any effect on this problem, but I can’t say for sure since I’ve never worked with them to the extent needed to find any bugs in the current browser versions.
I don’t know why (other than the possibility of the shadow roots affecting things) Kee doesn’t find these form fields. Looking in your debug log file might give some clues.
Unless Kee finds potentially relevant form fields to fill in, it won’t even look for entries matching the page URL. Standard type=password fields should be detected just fine though, hence my suspicion that this might relate to those shadow roots.
Is this website (or a demo of it) available somewhere public that I could access?