No, it doesn't. There only communication is happening through the site. The site issues a challenge to the PC, the previously registered phone confirms to the site that the challenge is met, and from now on the site trusts the PC. The PC and the phone can be on different continents.
The problem though is that you have to do this for every single site you access. So if you have 100 log ins and are switching PC or phone, you'll have to do this same dance 100 times in the next period. And of course, if you're switching because you lost your one device that was registered this way...
Edit: everything I say below is not just wrong, but confidently wrong...
"Communicate over bluetooth" doesn't mean anything. What app or BT device would they be using? How would a PC communicate with a YubiKey over bluetooth?
I have no idea where you got this strange concept from, but registering multiple passkeys from multiple devices on the same account on a site requires no communication between the devices - it only requires a trusted device to approve the request.
That's how it works. You open Google Chrome on Linux, press "Log in with PassKey", scan QR with iPhone, then iPhone contacts Google Chrome via bluetooth to do its crypto magic (which doesn't work 50% of times) and may be it'll work.
The problem though is that you have to do this for every single site you access. So if you have 100 log ins and are switching PC or phone, you'll have to do this same dance 100 times in the next period. And of course, if you're switching because you lost your one device that was registered this way...