Wow, uninstalling/reinstalling the app (via TestFlight) actually worked for me too! Thanks @Mosabon
initially it worked fine (after re-installing the app), after a few weeks it stopped working again, and I had to re-install the app again
quite annoying actually
I’m finding that FaceID doesn’t work on the first unlock, even if the main app has unlocked at least once with it. Is this due to the extension and the app running under different app identities and hence unable to use the same SEP unlock policies? Anyway, the point is that it’s frustrating :)
Updating IOS from 16.1 to 16.2 resolved the issue. I can now unlock the Safari extension with FaceID.
FWIW I removed the beta and installed the App Store version and now FaceID unlock is working again