If you're using an electronic id card then you don't have the same username and password values that your website uses. You likely won't even use the same fields. Authenticating through that usually just involves a separate button.
Auth systems don't key off of usernames... When we added FIDO support there's still a backing user store that records user x has FIDO y, with a recording of the public key. That's how the pairing works. You can see it today on login.live.com
