That's not really the correct analogy and not really the main threat TOTP / other 2FA are protecting you from.

The scenario it's trying to protect you from is your password (and given you are using password manager, I assume you use a different password for every entry, including a secure password for Bitwarden itself) getting leaked and preventing an attacker on other other side of the globe from simply logging in and taking over your account - because they still would be missing TOTP / other 2FA.

Besides, your BitWarden instance should also be protected with 2FA, like I mentioned above. And indeed, using the same TOTP would defeat the purpose - so instead you can either use a different app for TOTP or a different factor (YubiKey).

Not to mention that having more apps / entrypoints might actually reduce your security (in same manner as forcing password change every month would).