Briefly stated a "passkey" is stored locally, on the device. If you want to use a passkey, it will need to be enabled on each device that uses it. Some browsers incorporate it, some websites incorporate it (for certain browsers), and some companies are finally getting on board.
If you start using passkeys had you better still remember all your passwords?
Ideally, no, and it's an impracticable requirement different services tend to have wildly different password requirements. Case in point, one bank I used to do business with considered Safari's "secure password generator" insufficiently secure, and forced me to use what was to them a more secure password. That password?
"Password01"
That was ok. I wish I were joking. Needless to say I took my business elsewhere.
Passkeys are supposed to make passwords obsolete. That's a laudable goal, but it requires that the device itself is capable of some kind of secure, usually biometric authorization. For Macs, that's Touch ID or other means such as Unlock your Mac with Apple Watch - Apple Support. For iOS, it's Face ID. Theoretically — and this is something Steve Jobs advocated long ago — once you're in, you're in, and there shall henceforth be no more harassments for passwords or 2FA or other insults.
However, as you know, there are circumstances in which you will be asked for a passcode or password anyway. Apple is slowly, inexorably, moving away from that requirement... as it should be.
As far as security goes, passwords are horrible, and horribly outdated.
Boring video: Meet passkeys - WWDC22 - Videos - Apple Developer