Apple does not issue a password for your private emails. You only have a password for your AppleID and that never is shared with other companies.
Private relay isn’t technically the same as Sign in with Apple or Hide My Email other than they all are bundled with iCloud+
If you create an account with another service, it’s up to that service to integrate with passcode via Sign in with Apple where your device will use your main iCloud credentials to unlock a Pass Key token stored on your device and send that token (which is not your AppleID or iCloud password) to log you in.
If that service (chatGPT in your case) forces you to choose a password, it’s likely you used Hide My Email only and whatever you chose to send to them for them to store and authenticate. Again, this isn’t your AppleID password unless you reused it.
If you aren’t sure which path the service let you choose, check with chatGPT customer support. Or you could try their password reset procedure which typically emails a reset link or uses SMS / two factor auth.
You absolutely won’t use iforgot.apple.com in any case where you have a private alias or passcode sent to a third party. Only use “iforgot” if you can’t log in to https://appleid.apple.com with your Apple credentials.