The standard is called TOTP and Google became synonymous with it because they pushed it in the late 2000s for Gmail and have a large user base. Other sites did have systems beforehand, like Paypay which had a dedicated fob, but that was not widely used. Gmail was likely most people's first experience with MFA and Google pushed their own Authenticator app (and didn't really advertise that others could be used). As other sites got on board, it was easier to tell people to use the app they use for google to get their code, since you could assume people had the app.

Basically that made a situation where people who had a different TOTP app knew their app would work with "Google Authenticator" but for those without an app or using Google Authenticator, they were likely unaware of the interoperability and standards behind the mechanisms.

Not really, my guess is that you can import your 2FA codes from Google Auth so it is mentioned as an alternative to it