Why? I hope they don't, as I'm relying on my password manager to emulate a hardware token so I can finally log in to websites without needing a username/password.
At its core, FIDO2 is an authentication API, so the site can ask your browser to authenticate you (in whatever way the browser wants). If that's "talk to the password manager to authenticate the user using some fancy cryptography", why does the authenticating site care?
I'm looking forward to the day when my password manager only has one credential in it, my soft-FIDO2 private key.