n8726258 Great that you included a video of what you're doing when this happens; can now provide an answer immediately:
Play services does not support PIN entry over NFC, only when using USB. This is the case on stock Pixel OS as well. I know that's probably hard to believe, especially because you'd expect this to be a basic feature, and the dialog does give you an NFC option. (Not all keys are set up to require you to enter a PIN upon every authentication to a service, and services can also require you to enter a PIN independenly of that; so probably the dialog shows the NFC option just in case you won't need to enter a PIN. That's a very simplified explanation of how FIDO authentication works; technical explanation is available in the FIDO Alliance's CTAP specification documents).
You can instead use another app which supports PIN over NFC. It's not always reliable but it should work in this case and most other cases: https://codeberg.org/s1m/hw-fido2-provider
After installing it you can enable it in Settings app > Passwords, passkeys & accounts. As long as it's enabled there it should take precedence over the Play services dialog.