Is it possible to recover Google authenticator database
Google Authenticator uses "seed" values which are symmetric keys used to generate OTPs (One Time Passwords). This means there is more than one copy of these keys, one with Google and other with the registered applications. In a sense they are in pair, hence symmetric. When you try to register your new device with authenticator, there are more than two copies of these keys and this causes an authentication problem. If these keys were easy to be backed up and restored, the security might have been at risk. So, all you can do is re-register your new instance of Google Authenticator. For this you need to follow these steps-


Log in to your Google account (Since the Authenticator's key we have wouldn't be useful here, you can choose either Call or Message to get a code on your registered mobile number.)
After logging in, go to My Account > Sign-in &amp; Security. In the password and sign-in method click on 2-step verification (you will be asked to enter your password again).
On the next screen, you will see an option for Authenticator app, click Change phone and select Android or iPhone(accordingly). 
On your new phone install Google Authenticator app and open it, go to Set up an account and choose Scan barcode. Delete the previous entry if you have pre-installed the app and have an account registered.
On the next screen, enter the 6-digit key which you get on your app.
Let me know if there's an issue.