Do I need a separate Apple ID if I want to backup two computers

You cannot backup a computer to iCloud.

You can only sync certain data like Documents & Desktop, Notes, Contact , Mail and Calendar.

If both computers are using the same Apple ID for iCloud, then it will sync the data for both and merge it together, so you will have the same data on both computers and iCloud.

If you don't want this to happen, then yes, you need 2 different Apple IDs, one for each computer, specially if each computer belongs to a different person. Each person should have their own Apple ID, logged in to their own devices and should not share one with someone else.