How do I add to the 'allow in background' list?

It is usually up to the app to prompt you to give permission for the app to add itself to the list. I'm not sure how you would go about manually adding it (if that is even possible) since it may be a helper app that gets added & not the main app.

Check with the developer of the app for assistance if their app is not showing in the "Background" section where you can toggle it on/off.

Should that backup software be Carbon Copy Cloner ?

https://support.bombich.com/hc/en-us/articles/20686443528727-Granting-Full-Disk-Access-to-CCC-and-its-helper-tool

Otherwise, you would need to contact the Developer of this unknown backup software Developer

When you run the iDrive app (or maybe installer), then it should place the item into the Background Apps area assuming you did not deny it (I would think it should still show up but as disabled,but you never know with some developers). I guess you can try reinstalling the iDrive app, or make sure it is completely up to date & fully compatible with the version of macOS you are using. Otherwise, you will need to contact iDrive for assistance.

FYI, here is a FAQ from the iDrive website regarding your question:

https://www.idrive.com/online-backup-mac-faq#ventura

Edit: If you are running any anti-virus apps, cleaning/optimizer apps, or third party security software, then perhaps they are interfering with iDrive from working correctly. Those types of apps are not needed on a Mac anyway because they usually cause more problems than they solve, plus they interfere with the normal operation of macOS. If you have any of those types of apps installed, then uninstall them by following the developers' instructions.