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Since both this and disconnecting USB power altogether fixed the issue, I'd say that in some setups you have a ground loop (multiple paths to earth ground with different impedance). Ground loops can create electrical noise which interferes with sensitive things like touch ID sensors or makes audible noise in audio circuits.

(My hypothesis is that swapping from the 3 pin AC extender to the 2 pin plug helped by removing a ground connection and thus broke the ground loop - 3 pin is grounded, 2 pin is not.)

Take a look at all the things that might be electrically connected to the MacBook when you were experiencing this issue. How might they all be connected to the ground of an electrical outlet? Are there multiple outlets involved? Multiple power strips? Generally speaking, you avoid ground loops through practices like making sure that all AC-powered peripherals attached to a computer (and the computer itself) get power directly from the same power strip.
You are correct, this is something to do with the earthing / leakage current.

Here some improvised solution.

While using Touch ID ensure you touch the MacBook metal chassis the Touch ID responds.

I switched back to the three PIN extension cord haven't given much time on the two PIN OEM as my mains plug point is far away from where I work. Haven't been using the surge protection power strip either. Only scenario left is to use the two pin plug and check.
 
Thank you so much, this worked for me too on my 2021 16" MBP... was driving me nuts not being able to use Touch ID while plugged in...
can you please try touching the metal part of the MacBook Air while using Touch ID. For me it is consistently working fine (without removing the USB power cable). Appreciate your update
 
can you please try touching the metal part of the MacBook Air while using Touch ID. For me it is consistently working fine (without removing the USB power cable). Appreciate your update
I have 2021 16" MacBook Pro not the Air, but my reply to him was about that. I use MagSafe not USB cable but resting my palm is what works. I rest my palm on the metal part to the right of the trackpad and place my finger on the Touch ID sensor and it works, every time.
 
For me Touch ID also stopped working after about a week of using my 2021 16" MBP. So I tried to delete the finger print an re-add it. But when I try to re-add it and enter my computer password, I get this error:

Unable to complete Touch ID enrollment. Please go back and try again.

Hitting Try Again button, shows the same error!
 

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