Is it possible to restore Google Authenticator on another phone?

Unless you have previously backed up the QR code/secret key while initially setting up your Google Authenticator, this is not possible. Otherwise, you can contact each account's support team for help.

Alternatively, you can use another Authentication app like Authy (iOS and Android) or TOTP Authenticator (iOS and Android). These apps provide the feature to securely backup your codes and account details, so you can switch phone easily.