Is there any loss of privacy while using 2 Factor authentication?
If you use Google Authenticator on an android phone (where essentially you have linked your google id), to authenticate the login of a third party website, can google link your google id with the credentials of the third party website?
I don't think Google would ever risk their reputation to be destroyed in an instant by breaching your privacy that way... I have used Google Authenticator extensively for a good couple of years now, never had a problem at all.
When you activate 2FA on websites you are given a UNIQUE key together with a QR code to scan. The key works on ANY phone, so it is definitely NOT linked to your Google ID and you should therefore keep that key well hidden.