Edit: Maybe OpenSSH does offer resident keys. On the one hand their release notes say they do but on the other hand I was 100% sure somebody who should know insisted they didn't. A trawl of my records cannot find such a communication so perhaps I dreamt it. If resident keys are an option then you need to make sure to buy FIDO2 authenticators and to explicitly tell OpenSSH you want resident keys.

The current SSH FIDO behaviour doesn't do resident keys AFAIK. The OpenSSH team has discussed it, but not as I understand it written an implementation.

In a PIV setup Jim, Sarah and Gary can each have a device (say a Yubikey) with their own private key inside it. Both their workstations and shared desktop/ interactive servers can use these keys, when they're plugged in. So when Jim is sat at this workstation, Jim's key uses Jim's private key to authenticate Jim over SSH. If Sarah wants to use a machine she's never previously used, but it was set up for Jim and Gary, her key, Sarah's key, just works to authenticate as Sarah over SSH. Simple and easy to think about.

With FIDO non-resident keys a similar team Beth, Larry and Zoe have personal FIDO authenticators, let's say from Titan, they each set up their personal laptop to require their personal FIDO authenticator. Both elements are needed to authenticate. Beth's laptop, plus Beth's Titan key is enough to authenticate via SSH. But if Beth is at a Workstation she's never used before she can't use her Titan key to authenticate, it has no idea how to help on its own. If Beth sets that workstation up (a multi-step process) this isn't re-usable, Zoe can't use her Titan key without also going through that enrolment process.

The reason why is a mixture of differences between typical Web authentication journeys and SSH, and the deliberate (privacy preserving) feature paucity of FIDO Security Keys when used without resident keys.

The FIDO authenticator genuinely has no idea what your private key is (this is unlike for resident keys like Apple's recent announcement). It first needs to be presented with an ID, a large random-looking byte string. It has in fact encrypted your private key using AEAD mode with a secret symmetric key that's baked inside it and then used that as the ID. So when the ID comes back it decrypts it, the AEAD checks out, it now has a private key and can use that to authenticate you before it forgets it again.

On the Web in this case (again not the resident case like Apple's cool Safari feature) the ID is being delivered by a remote server to your web browser when you enter a username and the browser passes it to the FIDO authenticator. But in SSH authentication doesn't work that way.

So, OpenSSH stores the ID locally on a computer. It isn't secret, so it's not a huge deal if someone somehow steals it. But without that ID the authenticator can't do its job.

Maybe you could arrange to synchronise the IDs across machines in an organisation. But AFAIK nothing exists to sort that out today. So without either resident credentials (which need a more expensive FIDO2 authenticator) or some further synchronisation framework PIV is easier today if you want employees to authenticate from machines that aren't "their" personal machine.