Bitwarden Authenticator is a standalone app that is available for everyone, even non-Bitwarden customers.
In its current release, Bitwarden Authenticator generates time-based one-time passwords (TOTP) for users who want to add an extra layer of 2FA security to their logins.
There is a comprehensive roadmap planned with additional functionality.
No, it’s a separate app that is completely detached from the BitWarden App and servers.
More comparable to 2Fas, Aegis or similar apps.
EDIT: the fact that TOTP is not available on the free tier always irritated me. I am happy VaultWarden has it included, though it’s only one of three places I keep my TOTPs
You shouldn't keep your TOTP and passwords in the same place. All your eggs in one basket kinda thing. If your account is ever compromised somehow then you've lost everything.