It is expected behavior when Aurora cluster is restored from AWS Backup console. AWS Backup restores your Aurora cluster; it does not create or attach an Amazon RDS instance to your cluster. This behavior is similar to the restore-db-cluster-from-snapshot command [1]. As, this action only restores the DB cluster, not the DB instances for that DB cluster. You must invoke the CreateDBInstance action to create DB instances for the restored DB cluster, specifying the identifier of the restored DB cluster in DBClusterIdentifier. To create the DB instance you have to run the following command as discussed in Step 14 of our documentation in [2].

For Linux, macOS, or Unix:

aws rds create-db-instance --db-instance-identifier sample-instance \

          --db-cluster-identifier sample-cluster --engine aurora-mysql --db-instance-class db.r4.large

Further, you can restore the Aurora Cluster from RDS console, which provides the option to create the DB instance when restoring the Cluster. The backups taken via AWS Backup service will also be listed in the RDS console, and you can select the snapshot and restore with the required options.

There is a feature request open for this already, However, at this point, I am unable to provide an ETA if or when the product team incorporates this feature in their pipeline going forward. However, please refer to these links [3,4] for our latest updates and feature releases.

References: [1]restore-db-cluster-from-snapshot - https://docs.aws.amazon.com/cli/latest/reference/rds/restore-db-cluster-from-snapshot.html
[2] Use the AWS Backup Console to restore Aurora recovery points https://docs.aws.amazon.com/aws-backup/latest/devguide/restoring-aur.html#aur-restore-console
[3] Featured Announcements - https://aws.amazon.com/new/
[4] AWS News Blog - https://aws.amazon.com/blogs/aws/tag/announcements/​