It basically performs the same function as an SSH key (providing public key authentication), yes.

Your issue with logging in on your phone vs laptop can be solved by either syncing them (like the OS/Browser platforms of Google/Apple/Microsoft or a password manager like Proton Pass/Bitwarden do) or by setting up each device separately (like most people should do with SSH keys). Each method comes with trade-offs: syncing means they aren't device bound and can potentially be stolen, setting it up on each device can be a pain, etc.

The important thing to remember is that passkeys don't need to be the only authentication methods attached to an account. You can use the convenience of a passkey most of the time when it's possible and then fall back to another method (like a password/TOTP pair) when that's not available (such as when setting up a new device). There's also always the standard account recovery options if all else fails, those don't necessarily go away.

The other thing to remember is that it's not trying to be a perfectly secure solution to all authentication everywhere but to replace passwords with something better. Not having to generate and store random passwords with arbitrary complexity requirements, being able to log in with just a tap or a click, and not having anything that needs to be kept secret on the website's side can be enough of an improvement over passwords to make the change worthwhile.

You setup passkeys for all your devices with biometric features. I know I have a Yubikey for my desktop, facial recognition on my phone, and a fingerprint reader on my laptop. So, I setup 3 passkeys using biometric (fingerprint or face). I also kept my password and 2FA for now because it’s all new. I wouldn’t recommend jumping in face first.

I only am using it on a few key sites and partly because I’m a web developer testing it all out. I wouldn’t advise it for the average user at the moment but it’ll mature and many password managers can store passkeys now. As it matures, I’m hopeful it becomes seamless like FaceID and fingerprint readers.

Close, but you are still trusting the device you own. If I were to compromise that device, I could capture that key and use it. Again, this is my limited understanding, but a zero trust solution works in such a way that the actual keys are not stored anywhere. During setup, new temporary keys are generated. A keypass binds to the temporary key for use of authentication. The temporary key can be revoked at any time for any reason, whether it's due to a breach or routine policies. It can be as aggressive as it needs, and the implication is that if someone else (either you or an attacker) got issued a new temporary key then the other would not receive it. Using an incorrect temporary key would force an initialization again, using the actual keys that aren't stored anywhere.

The initialization process should be done in a high trust environment, ideally in person with many forms of vetting. But obviously this doesn't take place online, so there is the risk that your device is not trusted. This is why the process falls back on other established processes, like 2FA, biometrics, or using another trusted device. How this is done is up to the organization and not too important.

But don't get too hooked on the nuances of passwords, keys, passkeys,etc. The entire purpose is to limit trust, so that if any part of the process is compromised, there is nothing of value to share.

Disclosure: Worked in military and this seems to be a consumer implementation of public/private key systems using vector set algorithms that generate session keys, but without the specialized hardware. It's obviously different, but has a lot of parallels, the idea in this case is that the hardware binds to the private/public keys and generates temporary session keys to each unique device it communicates with, and all devices can talk with members of it's own vector set. Capturing a session key is useless as it's constantly being updated, and the actual keys are stored on a loading device (which is subsequently destroyed afterwards, ensuring the actual key doesn't exist anywhere and is non recoverable, but that's another thing altogether). My understanding of passkey systems is solely based on this observation, and I have not actually implemented such a solution myself.