Touch ID with Magic Keyboard not working and Unable to add a new fingerprint

RetiredSafety wrote:

Updated my 2021 iMac system to Ventura 13.3.1 last week and can no longer use Touch ID on the Magic Keyboard for my iMac. Cannot use any Touch ID functions to access passwords, use Apple Pay...a total loss of Touch ID functionality! Two Apple support articles ("Use Touch ID on Mac" and "If Touch ID isn't working on your Mac") have not resolved the problem. Any others with the same problem? Any solutions?

iMac external keyboard…

switching off the keyboard for 30 secs or so and then hard connect the keyboard via lightning (charging) cable. Then switching on the keyboard, try again your TouchID set up.

if no resolve try a SafeBoot How to use safe mode on your Mac - Apple Support will sort many anomalies

With a wireless keyboard with rechargeable batteries connect the charging cable before booting to Safe Mode. 

This will make it act as a wired keyboard and will result in a successful Safe Boot mode. login and test, reboot as normal and compare your results.