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?
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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.