Shortly after posting my last comment here, I found a suggestion to reset SMC (System Management Controller) on my MBP M2 (2023). All articles says MBP M2 does not have SMC or it resets when you restart your MacBook. Not sure if that's true, the steps helped! I can add a finger print and it's working!!! We'll see for how long, but at least it's a start.
Followed these steps:
Shut Down Your MacBook Pro:
First, shut down your MacBook Pro completely. Go to the Apple menu and select "Shut Down".
Connect to Power:
Make sure your MacBook Pro is connected to power using the AC adapter.
Perform the SMC Reset:
Press and hold the Control (left side) + Option (left side) + Shift (right side) keys on your keyboard.
While holding these keys, press and hold the power button as well.
Keep all four keys pressed down for 10 seconds.
Release all keys and the power button at the same time.
Turn on Your MacBook Pro:
After releasing the keys, wait a few seconds.
Then, press the power button to turn on your MacBook Pro.
EDIT: Actually, I take it back. After restart Touch ID cannot be used and a new one enrolled. However I do have a fingerprint saved now, so it's a step forward. I know it's must be a system issue. I was able to use it once. I expect to see a similar behaviour now - so from time to time it'll allow me to use it.