I’m stealing this from another comment:
The main advantage comes with phishing resistance. Standard MFA (time based codes) is not phishing resistant. Users can be social engineered into giving up a password and MFA token. Other MFA types, such as pop up notifications, are susceptible to MFA fatigue. Similar to YubiKeys, Passkeys implement a phishing resistant MFA by storing an encryption key, along with requiring a biometric. The benefit here is that these are far easier for the average user, and the user does not need to carry a physical device. Sure, fingerprints could possibly be grabbed with physical presence, but there is far less risk that a users fingerprint is stolen, than a user being social engineered over the phone into giving creds. For most organizations and users, this is far more secure.
You don’t have to store them with Google. Passkeys are supported in both iOS and Android natively. Within the last few months both Bitwarden and 1Password support storing passkeys as well.