If you haven't already tried, you may want to toggle "Use Touch ID to unlock your Mac" in Touch ID and Passcode. Sometimes these unlock settings say they are enabled but they have something underlying broken and toggling the setting off then on fixes it. I've seen this behavior with using Touch ID for the App Store, for example. It kept prompting for password until the setting was toggled.
Also important to know if you've upgraded to macOS Ventura. The new System Settings appears to have taken away our ability to check existing registered prints by unlocking Touch ID and password, then pressing the finger you want to test against the registered prints. If registered, the print used to turn red but now there is no reaction, even when everything else with Touch ID is working perfectly.