@kreesher Welcome to the forum!
There is a free Bitwarden Authenticator app for iOS and Android.
Thanks, But this app don’t sync between devices. So currently is not a full solution
This is now possible. The standalone authenticator app can now sync with your bitwarden password manager, which does sync between devices.
This is bad idea. Storing ones Bitwarden TOTP solely inside Bitwarden is a very good way of locking oneself out of their vault. The problem being that one needs to be logged in on at least one of their devices to access the TOTP to login on their other devices. But what does one do if logged out of all their devices?
There is nothing wrong with storing a copy of the TOTP secret in the Bitwarden vault; the problem is with storing the only copy.
And, if also stored elsewhere, there is no need for it to be visible in (for free) in the Bitwarden Password Manager vault.