I didn't understand the question as written. 2FA comes in many flavors, and I'd (incorrectly) assumed you were referring to the site sending a code via email or text message, which doesn't really care about the email address in use, as long as you get the resulting email, and that's the point I was trying to make.
However, it's clear you're now referring to the rolling validation codes model used by Passwords.app, as well as other apps such as Google Authenticator and Microsoft Authenticator. These don't require the site to send you a code, it's synced with the app on your Mac/Phone.
Ultimately, it's up to the site in question to determine which 2FA models they support. Relatively few sites support the validation code model, opting, instead, for the SMS or email verification.
So I think the answer to your question depends on the site you're accessing, and the 2FA model they use. Personally, I don't see an option to use 2FA with ChatGPT, but that may be because it's relying on my Apple Account for authentication.
If a site does support validation codes, you can set them up in Passwords.app via the Codes button:

Under Codes, there's a + button to add a new verification code, which will need either a setup code or a QR code from an app/site that supports this feature.