Q: Is the password used to backup iPhone unique?

Are you referring to an iTunes or an iCloud backup?

GB

OK, so if it is asking for a password, it means the backup is encrypted, so you need the encryption password. If you don't know what it is, then you need to try any password you can think of. Your Mac Admin password, Apple ID password, Lock Screen password, anything that you may have used and put in here as a password. You can try as many times as you wish until you find the password.

If you cannot find it, then you will not be able to use the backup.

Best of luck,

GB

How about using Gail's Magic Solution:

To bypass the issue of encrypted backups always being encrypted:

Backup to iCloud. Then erase the phone. Then backup to iTunes as an unencrypted backup. Then restore from the iCloud backup. Then backup to iTunes again.

From: https://discussions.apple.com/message/30209896?tstart=0#30209896

Thanks elcpu! Unfortunately, it sounds like the OP has already erased the device

GB

About encrypted backups in iTunes - Apple Support

If you use a Mac to backup your iPhone, there is also the potential that you checked the box to allow Keychain Access to keep this password for you as well. You can open the Keychain Access application on your Mac and do a search for "iPhone Backup." Double click on this and click where it says Show Password. You will be prompted for your Mac Admin password to your computer to reveal this.