Personally, I don't, I explicitly and intentionally keep my Email, Password Manager and 2FA providers all distinct and separate. (This was very useful for extracting the TOTP secret out of the Symantec VIP 2FA system: https://github.com/dlenski/python-vipaccess)

At work, we do for non-person accounts. It's non-ideal, but I don't want to manage two separate systems for storing that data and trying to keep them in sync. The password vault system itself has a hard requirement for 2FA to download the vault.