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.