Your both right and both wrong. Biometric authentication is an identity scheme. The combination of username and password is also an identity scheme. A certificate chain is another identity scheme.

Identities both identify who you are and are, ideally, difficult to fake. Username password artificially handled those two concerns separately, but that doesn't mean that all identity schemes must do so. For them to say it's your password is wrong, but for you to say it's your username is also wrong. It can be thought of as both or neither but it isn't either one on its own.