Yes, you can. Either you can physically own the key on the physical device (i.e. Yubikey/Google Titan/HSM) or you can use software (KeepassXC, Bitwarden).

Most operating systems and browsers come with a sync mechanism that many people default to, but it's no more than that: the default.

As for account recovery, most websites have a way to recover your account when you lose your password, there's no reason why that wouldn't work for passkeys. Every website with passkey access I've used so far makes passkeys optional and forces you to set a password already. If they switch their default to passkeys and add a password as an optional step, nothing would change.