What would be the suitable solution in AWS backup for our use case? -> AWS Backup will completely fit your use case. You can add a particular tag (key-value pairs) to your RDS and EBS resources.
Can we call a backup job from our script using CLI and make sure the job will be executed immediately? -> There is no difference between the backup job executed by the backup plan or on-demand backup using CLI. The backup job will be in running state with start within the window mentioned in the backup plan or CLI command. [+] https://docs.aws.amazon.com/cli/latest/reference/backup/start-backup-job.html
In our use case, do we need to create a job per DB or there is an option to create something generic to all but still have flexibility to trigger the backup from the script? -> You can create a single backup plan for all the RDS and EBS resources with tag based selection and associate same tag to each resource if the retention, start time, backup vault is same. You can also create different backup plan if the start time, backup vault, retention or tags associated with resources are different.
-> So, the reason you should switch from script to AWS backup is that this will eliminate the use of scripts, and it's maintenance + there is no extra charge for AWS backup service as you pay for snapshot, and you have benefits of AWS backup like Vault lock, backup protection etc. [+] https://docs.aws.amazon.com/aws-backup/latest/devguide/whatisbackup.html#features-for-all-resources
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