How to set Touch ID fingerprint on Mac mini M4 with Sequoia 15.3 using a Touch ID keyboard

Barney-15E did suggest plugging in the keyboard…

"If you don’t have touch ID settings, plug in the keyboard with the cable and see if it recognizes it. If it does not go to the Bluetooth settings and forget the keyboard, then re-add it."

From the manual: If Touch ID isn't working on Mac - Apple Support

"Turn the keyboard off and back on. Then use a USB cable to connect it directly to your Mac. Wait at least 10 seconds before attempting to set up or use Touch ID again."

I don’t know what you referencing regarding double tapping the fingerprint ID button, but you open the touch ID settings in the system settings to set a fingerprint

if you don’t have touch ID settings, plug in the keyboard with the cable and see if it recognizes it. If it does not go to the Bluetooth settings and forget the keyboard, then re-add it.

and now that I think about it, are you using an apple keyboard with touch ID?

Perhaps this will help: Use Touch ID on Mac - Apple Support

FWIW, the power button on a Mac mini has nothing to do with setting up Touch ID.