> neither provider is asking to sync your private key.

Yes, they are. According to the white paper linked in the press release:

Just like password managers do with passwords, the underlying OS platform will “sync” the cryptographic keys that belong to a FIDO credential from device to device.

https://media.fidoalliance.org/wp-content/uploads/2022/03/Ho...

Ars Technica had a better write-up of these announcements back in March: https://arstechnica.com/information-technology/2022/03/a-big...