How do I create a back up for my child in family storage?
How do I back up my child Mac air in family storage in iCloud?
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Apple offers iCloud Backup for iPhones and iPads but not for Mac computers.
It is possible to sync data from a Mac to iCloud if you turn on those options (e.g., iCloud Photos, iCloud Drive including the Desktop & Documents option). Syncing has some of the benefits of a backup (and the benefit of being offsite) but is not the same as a backup. Major differences include the fact that not everything on your Mac can be synced and the fact that if you delete something, accidentally or otherwise, there is only a limited time period (30 days in most cases) during which it can be recovered.
Also see: How to back up your Mac - Official Apple Support
And more information on syncing in:
Set up and use iCloud Photos - Apple Support
and
iCloud Drive FAQ - Apple Support
If your child is sharing family storage, he or she still uses their own Apple ID and they would sync to iCloud the same way as if they were not sharing storage. Also see: Share an iCloud storage plan with your family - Apple Support