Thanks.
I can finally report success. 😃
Did try everything I could think of, really, and more than just once:
* remove the keyboard from Bluetooth devices in System Preferences altogether and re-pair (to work as a standard keyboard, without Touch ID)
* then tried to pair the Touch ID sensor ...a couple of times
* re-started my Mac three times
* installed the macOS 11.5 upgrade in between (though I was on 11.4 before, which should have worked and did recognise Touch ID in System Preferences. But didn't on the first attempt at pairing Touch ID post-upgrade)
* connect the cable via USB to Lightning cable
I'm not sure what did the trick, in the end. One of those things where you aren't really sure what exactly did the trick - other than just luck. If anything, it may have been disconnecting the Lightning cable again: the keyboard seems to being a USB when the keyboard is connected to the Mac via its Lightning port - upon unplugging, the Bluetooth "Connected" overlay notification comes up again, suggesting that it only then switches to Bluetooth for its connection again. Though I don't think that a USB connection should prevent it from pairing Touch ID and macOS certainly didn’t tell me so. Also, my very first two or three attempts had been without the cable as well.
Must have been the most attempts I've had to make at signing in to or pairing any (non-defective) Apple product or service ever. 🤨 Thankfully that shouldn't be necessary often, the standard keyboard functionality (without Touch ID) was very reliable and everything seems to be working fine and stable with Touch ID now. 👍