This is because Ubuntu 22.04 LTS ships all browsers using the "snap" system, which doesn't work with the eID, as explained on https://eid.belgium.be/en/faq/why-it-not-possible-use-eid-software-snap-andor-flatpak#7636

The solution is simple: use a browser that is not shipped using snap. The above FAQ entry gives a few download links to browser versions that will work.