I set up Google 2 Factor Auth to log into other sites. I see there are backup codes if you lose your device.
Are these used only to log into Google, or can they be used to log into other sites?
The backup codes provided by Google will only work for Google's services.
The Google Authenticator works on the standard HOTP and TOTP algorithms described in their RFCs. Google just provided a convenient app to use it. HOTP and TOTP are relatively easy to implement algorithms and most popular languages has libraries supporting it. In fact, here is one I wrote for .NET.
It is up to each individual site implementing the HOTP or TOTP algorithms through Google Authenticator to provide a means to recover from the loss of said Authenticator.