iCloud

You should each have your own iCloud account.

They can be linked under a Family Sharing account so that music, photos, storage, etc, can be shared between the accounts, but with a degree of separation so that emails, messages, etc. are distinct.

https://www.apple.com/family-sharing/

Apple ID's should be a personal thing and not dependant on devices. i.e. if Mr NWW has a phone, iPad and computer, he should use his Apple ID for signing into all the services he wants to use on those devices (iCloud, iTunes, messages, books etc) if Mrs NWW has an iPhone and a Mac, then she should use her Apple ID on those devices for the services she wishes to use. If Mrs NWW also uses Mr NWW's Mac, then Mr NWW and Mrs NWW should each set up their own user account (separate user accounts are only available on computers, more about users at https://support.apple.com/en-gb/guide/mac-help/mtusr001/mac) each user should use their own Apple ID in their own user environment.

If Mr NWW used his Apple ID on a previous Mac, any services he enabled such as iCloud contacts, iCloud calendars, iTunes etc should also make that data available to his new Mac when he enables those services under the same Apple ID on it.

Think of it like this: the only account is the account you get with an Apple ID. You then use that Apple ID to sign into different services such as iCloud, iTunes, Messages, Books etc.

I'm guessing that what you have done is after setting up one Apple ID to use with iTunes you have them set up another Apple ID to use with iCloud, would this be correct?

If you both have your own separate Apple ID, then you can use it to log into iTunes, iCloud and all the rest. You should only have one Apple ID each. You will want to reuse the Apple ID that you have used to make your iTunes purchases on, because you cannot transfer those purchases to another Apple ID.

You cannot use iCloud to back up your computers. iCloud is great for backing up devices like phones and iPads, but a computer should be backed up to an external drive.

I am of course assuming that you really mean backup, iCloud will sync your calendars, photos, contacts etc to the cloud, and many people often refer to this as a backup, but it isn't, it's a synced set of data, and if you delete something in one place, it will get deleted in the other.

You should always have your own Apple ID's. I see too many folk here that have lost valuable data or photos because someone they have shared an ID with has deleted it.