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

Great addon!

KeePass lets you keep your own password vault on your own system, or sync using the service of your choice.

The browser extension takes a little effort to connect it to your desktop app the first time, but from that point on it mostly does its job and stays out of your way, detecting login forms and adding a button to fill in credentials you've stored with that URL in KeePassXC (When you have the desktop app open and the database unlocked).

Tips on matching websites to password entries:
1. General URLs will match better, like example.com instead of example.com/login?fromSignIn=true.
2. You can associate extra websites that use the same account by putting them in extra fields named KP2A_URL, KP2A_URL2, etc. in the desktop application. (Conveniently, these are the same fields KeePass2Android uses.)