No, Google Authenticator is not tied to a device. It's a standard (TOTP, RFC 6238) and you just need to use an app (perhaps not Google Authenticator, I use a different app myself) that will let you see the numerical code that you need to save somewhere.
It’s tied to the device as in it won’t be part of your backup to a new phone.. you have to manually transfer it yourself which according to you is using another app!
So the likelihood of moving to a new phone without those codes transferred is very high. Not exactly an easy experience.
