Yes, when you export Google authenticator codes from one phone to another phone, both phones should work the same. (Can verify by seeing that both phones generate same codes at same time).
I would recommend instead of Google Authenticator to use
Twilo Authy Authenticator because in addition to including the same features Google Authenticator already has (2FA, and the ability to use multiple devices at same time), it also has a backup function which means you can setup Authy to backup accounts securely to the cloud so you will NOT need to worry about losing any accounts/codes that are already associated with the app and if you need to install Authy with another phone, you can easily just sync everything over and be right where you need to be, even if you don't have the other phone!!
The Backup function is WHY I am using Authy instead of Google Authenticator. (If you uninstall Google Authenticator or lose your phone or whatever, there's no backup and can't recover anything associated lost phone or uninstalled app. Potentially locked out of everywhere you set up with Google Authenticator, except if you have backup codes or the site/service has a way to recover from lost 2FA devices.)
