Microsoft authenticator backup not saving

Question

I have a new phone and want to install my microsoft authenticator. The app is downloaded and I have made a backup in the authenticator on my old phone.

1. On my new phone I clicked on "Start restore".
2. Then I have to choose an account to log in. I have logged in to the account in which I also made the backup in my other phone.
3. After logging in the following text appears: "Cannot restore. The backup is not saved under ..(email address).. Log in to another personal microsoft account."

I have tried several times, but I always get the same thing which prevents me from installing the backup.

I would like to know what the problem is. Can this be solved? If so, how? :)
Or is there another way to easily transfer the passwords?

Thanks in advance!