I can register an authenticator multiple times, for instance to represent multiple different accounts of my own, or represent multiple people on a shared device.
If I delete a credential, the expectation is that registering a new credential is not going to correlate the authenticator (and thus the user).
If I want to have hygiene steps of rotating the cryptographic key a user uses to log in, I won't want registration to create the same key pair each registration.
And for the cloud sync:
The UX can present that web authentication is an option to log in. The user will be confused if that option is presented for sites which will not recognize the authenticator.
The site can store data alongside a credential to be returned to optimize the log in process, such as a site-specified identifier to look up the user credential in a database. That state needs to be synchronized.