There is no mechanism for that. The biometric hardware is local to your machine. I suppose the hardware could emit a "biometric fingerprint" that is transmitted to the server. And if anyone acquired your fingerprint they could unlock any site. Ok, so you could add some site specific salt to the process before sending the data. This is pretty close to just having a local password manager where the master key is biometric rather than a password.
But touch/faceID aren't implemented that way. They intentionally don't expose the biometric profile, that is kept local to the secure enclave. They instead just give you access to the secure enclaves keystore. Rather than signing data you could just use uniquely generated public keys like a password, or do something like signing a websites name with a private key to generate a password.
However, these approaches don't really make sense. The advantage of public-key cryptography is that you prove who you are WITHOUT SHARING the private/secret key. This is much more secure, because it prevents threat actors that don't have access to the private key from replicating the "proof" or signing process. This is what attestation is about. You can design alternative attestation schemes, but webauthn is pretty simple.
But touch/faceID aren't implemented that way. They intentionally don't expose the biometric profile, that is kept local to the secure enclave. They instead just give you access to the secure enclaves keystore. Rather than signing data you could just use uniquely generated public keys like a password, or do something like signing a websites name with a private key to generate a password.
However, these approaches don't really make sense. The advantage of public-key cryptography is that you prove who you are WITHOUT SHARING the private/secret key. This is much more secure, because it prevents threat actors that don't have access to the private key from replicating the "proof" or signing process. This is what attestation is about. You can design alternative attestation schemes, but webauthn is pretty simple.