That's the best way I can describe it on one line.
This is a 2021 M1 iMac with extended Magic Keyboard. Updated to Ventura 13.3.1 during this investigation, but started on Monterey 12.2 - no change in symptom.
For a time after boot [& initial regular typed login], the Touch ID button/sensor works perfectly, for passwords & also to lock and unlock. So long as the machine is attended & 'working' it continues to function as expected.
If left alone to sleep the screen [but not the Mac] - it seems to be the time designated as 'Require password nnn after sleep or screen saver begins" - then TouchID is no longer recognised.
The login screen at that point says "TouchID or Enter Password" but only the password will register. If left for even longer, the login screen changes to just 'Enter Password', as though no TouchID button was recognised.
I've realised that though this might require the time for 'sleep' the Mac rarely actually sleeps during the day - we have an internet radio running, which prevents it. So it's not the sleep process itself, but some other timeout.
This lack of recognition extends to using the same button to Lock the Mac. It simply no longer works; so it's not just the fingerprint sensor, but the entire key that's ignored/forgotten.
I have tested with & without safe boot, through to creating a new Admin account & trying to register a new fingerprint. The button is not recognised at all. After a reboot all is once more OK & I can finish the add fingerprint process. This new account then behaves as the original - with this strange 'timeout'.
I have a workaround which points to something, though I'm not certain quite what.
If I plug the keyboard into USB & leave it connected, after a while the TouchID is once more recognised & stays recognised so long as it's connected over USB rather than Bluetooth.
I can cycle between these behaviours repeatedly, without rebooting.
I'm left trying to diagnose if this is essentially a software or hardware issue, keyboard or Mac, without really knowing what to look at next. The Mac is quite 'clean' as far as non-standard & third party software is concerned. I ran EtreCheck & tidied out a couple of things we can live without, but there should be nothing that could get in the way of Bluetooth.
Any suggestions as to next approach?
Late Edit
No change on macOS 13.4
As there are a couple of confirmations of this behaviour in comments, I've submitted a bug report to Apple [12 June 2023].

