You share public keys when registering the passkey on a third party service, but for the portability of the keys to other password managers (what the article is about) the private ones do need to be transferred (that’s the whole point of making them portable).
I think the phishing concerns are about attackers using this new portability feature to get a user (via phishing / social engineering) to export/move their passkeys to the attacker’s store. The point is that portability shouldn’t be so user-friendly / transparent that it becomes exploitable.
That said, I don’t know if this new protocol makes things THAT easy to port (probably not?).
The “.net” makes me think it would be some online service that runs on remote servers or ties you to some online platform without any real gain other than just to increase their level of control over the customer.