What's supposed to happen is when you tell the site you want to use a passkey and one is not available to your Linux desktop's browser you are shown a QR code that you can scan on your phone. The login will then take place via the phone using your passkey that is on the phone for that site.

If you want to test to see if your browser handles this right you can do so at <https://www.passkeys.io/>.

Once you are logged in with your passkey from you phone you should be able to go to your account settings on the site and somewhere in there find an option to add another passkey. You can then add a passkey generated by your Linux browser or your Linux password manager if you use a password manager that supports passkeys.

Some will object that this is not good enough because they might want to login to some desktop they have never logged in from before when they do not have their phone handy.

That's probably not as big a problem as they expect though because unless you are using passwords you have memorized the same problem applies to passwords. I've got over 400 accounts in my password manager, almost all with long random unique passwords. That means I'm not going to be logging in somewhere new to any of those sites unless I've got access to my password manager, which in practice means unless I've got my phone or tablet with me.

FWIW I don't think that this makes passwords redundant in general, but with passkeys, password becomes a last-ditch safety valve to regain access to the account. Meaning that it can be generated, very long, and stored in a way that is optimized for safety and security over ease of access (like, say, an encrypted text file on multiple USB sticks stored in different physical locations).