This is my fundamental problem with passkeys: I don't want to use any syncing service.
To be clear, I don't want to deprive other people of the ability to sync their credentials; I simply want to opt out myself. I just want to be able to manually back up and restore my credentials, like I've always done with passwords, but the passkey vendors seem to want to refuse to give anyone this ability. The vendors claim that this is to make phishing impossible, but I abhor paternalism in all forms, and also it's suspicious that this paternalism forces people to use the syncing systems of the passkey vendors, which are usually paid subscriptions. So passkeys become an endless supply of money for the vendors.
It's very telling that passkeys were designed and shipped without any export/import mechanism. You can plainly see the priority of the passkey vendors, which is to lock you in. Allegedly, export/import is coming sometime in the future, but I strongly suspect that they'll end up with some kind of "approved provider" system so that the big passkey vendors can retain absolute control and avoid giving power to the users.
I haven't really looked into it myself, but it seems to be using the same database format as KeePass, and it hooks into macOS's "FIDO provider" API, which makes it accessible to not only Safari but all browsers that use it (which includes Firefox and Chrome on macOS, and probably everything on iOS), without requiring any browser-side extension.