When you enable two-factor authentication on websites, they usually show you a QR code and ask you to scan and launch your authenticator app. Some sites specifically ask you to use Google Authenticator, you don’t have to. What Google Authenticator uses are the HMAC-Based One-time Password (HOTP) and Time-based One-time Password (TOTP) algorithms.
Other authenticator apps like Authy, Duo Mobile, Lastpass, and 1Password all implement the same algorithms and are able to generate the exact same tokens you get from Google Authenticator.
The major difference among different authenticators is that some integrate with the cloud and upload an encrypted copy of your keys to their server, so you enjoy the convenience of accessing your tokens on multiple devices. Of course, in this case, you are also putting trust on the authenticator provider.