The password manager returns passwords directly. They can be viewed, memorized, passed to another person, copied to another device, or checked into git.
With TOTP, there is a private key inside, but it is not accessible to user. You cannot view it, or memorize it, nor can you pass it to another person or check it into git. It is purely implementation detail which is not exposed in any way.
Disclaimer: this is the case with classical TOTP devices, like RSA SecurID hardware token, or un-rooted Android phone running Google Authenticator. I have those, and everyone I know have them as well.
There are exceptions, like people using LastPass 2FA or people who store TOTP secret on their PC. This is not intended usage, and it does not matter for most users.