Here's what the Electronic Frontier Foundation says about fingerprinting.
"""Digital fingerprinting is the process where a remote site or service gathers little bits of information about a user's machine, and puts those pieces together to form a unique picture, or "fingerprint," of the user's device"""
From: https://ssd.eff.org/en/module/what-fingerprinting
Here what TeamViewer is doing isn't fingerprinting. it's not combining unrealated bits of information to uniquely identify a user/computer. it's looking at literally one bit of information to identify whether the current non-uniquely-known user is in a -large- group or not, the group of "computers with teamviewer installed".
it can be claimed that this is adding to the bits that can improve a third party's ability to uniquely identify a user/computer, but that's a different claim, that's not what teamviewer is doing.
The action of taking your fingerprint to identify you is fingerprinting. Providing you a handrail without a purpose of identifying you, even though it happens to take your fingerprint for anyone else, is not fingerprinting. Changing your fingerprint is not fingerprinting.
This is an abuse of a technology with more harm then benefit if you ask me. Calling it "fingerprinting" is still a category error.