A new ‘family’ member cannot backup their iPhone

You seem to be speaking of 2 different things.

The iPhone's local storage, and the shared iCloud Storage are 2 different things.

Having iCloud Storage does not make the iPhone have more local space.

If the iPhone's local storage is full, then it's full. iCloud will not affect this.

The family member would need to delete content off the iPhone to make room there.

For backups the family member can go into Settings ➜ Account ➜ iCloud.

If the iPhone itself is full, they can go to Settings ➜ General ➜ iPhone Storage and see what's taking up space.

click here ➜ Manage storage on iPhone - Apple Support

So what did that have to do with “the phone is full” then?

What happens when the family member goes to Settings ➜ Account ➜ iCloud ➜ Backup Now?

Does the family member see the correct iCloud storage in Settings ➜ Account ➜ iCloud?

Help us help you by being very clear about what is happening.