Symptoms

1. You run a backup of Microsoft 365 mailbox with Acronis True Image 2019/2020
2. Backup fails with the following error: 

   Cannot back up this mailbox because the login for the account to which the mailbox belongs is also used for another Microsoft account

Cause

There are two two types of Microsoft accounts: Personal account and Work or school account. If you have a Personal account and a Work or school account tied to the same e-mail address, Microsoft 365 mailbox systems provide incorrect replies to commands utilized for accessing data to back up.

Solution

You can use the following workaround:

1. Go to manage account page on https://account.live.com and create an alias email address, with a login different from the account that produces the error (click Manage your sign-in email or phone number -> Select Add email). 

You can find instructions on creating and managing Microsoft account aliases in this Microsoft Support article

2. Set the newly created alias as primary alias on your Microsoft account.

3. Delete the existing Microsoft 365 backup in Online Dashboard: on the "tile" with Microsoft 365 backup task click the "gear" icon and select Delete from the drop-down menu:

4. Use the newly created alias to add new Microsoft 365 for Home backup.