It is worked with Ubuntu 22.04.1 LTS now.
I just had to execute command:

flatpak permission-set webextensions org.keepassxc.keepassxc_browser snap.firefox yes

It would be fair to mention everywhere the well-known problem with snaps on Ubuntu that dramatically cripples the usability of this otherwise great software. Ubuntu is just the largest Linux distribution by far, and you've lost it...

As I said some years ago, you need an alternative way of communication. Shouldn't be so hard, Mozilla Docs says you can use sockets to communicate with local applications (by Native messaging). It's presumably just plain TCP/IP. Even password managers are mentioned as examples...

Time has come - already in 2021 but I postponed the upgrade so far -, and there is no apt-based version of FireFox as a second choice anymore in Ubuntu, only the Snap Store-based, so we are doomed. I will not go backward installing a browser from a place out of the Ubuntu's own ecosystem. Thank you.

---
My response to your response:

Thank you, but an alternative means an alternative. If not the Native Messaging then an encrypted file or similar would have been developed as a way of communication. We talk about years... No way that a FireFox extension can't and couldn't communicate with the outer world.

But as I'm reading here, see the link below that is really easy to find, no doubt, the problem not handling the Native Messaging correctly is in the browsers themselves, and it is already _solved_ after some years in FireFox (in 105..108 betas so far), KeePassXC is working (!) in the tests, I'm about to try it soon. So you are also not informed well enough for some reason...

https://forum.snapcraft.io/t/call-for-testing-native-messaging-support-in-the-firefox-snap/31055/36

It is only in the Beta channel until the next version of FireFox, probably since it is a fresh development from this summer, but it is OK to me because there is a snap of every FireFox beta as well in Ubuntu, and I can install it in parallel to the non-beta version for now if I want.

As I told you, Ubuntu decided sometimes earlier to go with only the snap version of browsers and even if I'm not hundred percent sure it is the best way to go forward, I will not install the less safe and unsupported apt version of browsers again.

Can you please tell me how to make it apply passwords to both password fields? Currently when I click on generate password on a password field it applies a password only on that field and when I click on generate password on the second field it a different password. Also when pressing on the add-on it just says "redetect password fields" which is why I have to do each field like described above.

Great addon!

Lässt sich nicht konfigurieren, Startet nicht. Findet die daerhaft benutzte Datenbank nicht. Keine Automatisierung möglich......

As reported by other users, and NOT EVER properly mitigated by supposed "gurus" of FireFox. . .here is the crux of the current issue. . ."connected" KeePassXC DBs are somehow NOT integrated with the browser (this has been verified on current versions of FireFox, FireFox/Tor, Edge, and Chrome); that is, the KeePass extention keeps displaying a small window that seemingly requires "Redetect login fields." HOWEVER, each and every attempt to on/select the field to do so, results is NOTHING (that is NO detection happens and user is continually in a "loop" of trying to and redetect, ad neaseum!) Clearly there are some nefarious "players" in the Windows 10 realm! MITIGATE PLEASE! and stop telling users to do QA-ing!

for no reasons the extension stopped connecting to my database and the troubleshooting is not helping solving the problem. it is a really a shame the security of the app is great