What do you mean by logging the code? On bitfinex 2fa settings page I can see a QR code and an "Accont token(key)" which is a string. Do you mean that I can for example print this page, and when I loose my phone next time I will download a google authenticator app on my new phone, feed it with the QR code that I printed and It will work as the old one?
Yes, it will work. That is the secret key that you set intro your phone in order to be able to generate the keys you use to login.