Yeah, I am having the same problem, Iphone 13 pro max. I updated from 16.0.3 to 16.1 and Face ID stopped working.
I tried the different tricks / links (If Face ID isn't working on your iPhone or iPad Pro - Apple Support), but it did not change anything.
I downgraded to 16.0.3 and it worked, for a while. Then it began to fail again. But it might have been just a fluke.
It actually works fine for 5 minutes (identification, setup alternative appearance), then it fails for minutes / hours in the same spot, same light settings.
I upgraded again to 16.1 and I have the same behavior.
When it fails in 16.0.03 I see the "Face ID is not available", in 16.1 I only see "put your iphone a bot lower / higher" messages.
When I try to setup the face with a mask, I have the message "Your face must match the existing FaceID appearance".
AT&T is sending me a new phone, so I'll see if I have the same problem with the new one.
You never realize how annoying typing passwords on phone are until you have to :)