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.
Available for iOS and Android
They’re probably using HOTP or something else, not TOTP. TOTP is literally just the key + any clock. Or maybe it’s the “click button to authenticate” and not the “enter code to authenticate,” which might not be HOTP or TOTP, but something else entirely (e.g. Steam’s system is neither AFAIK).
If it’s TOTP, you just need to get the key and can use any authenticator app.