The registration is because there exist clever amoral people.
If they didn't require you to register the warranty to a particular machine while that machine is still under its original warranty period you could buy one AppleCare and wait to see which of a number of machines broke, then register it for that one. Better yet, buy a bunch of AppleCares each month then sell them a year later on a secondary market to people with broken computers.
If Apple accepted a receipt as proof of co-purchase, then people would print receipts.
Maybe Apple could have a refund program for people that bought AppleCare and failed to activate it? Their computers must know how many AppleCares go unactivated. Perhaps it isn't a widespread problem. I suppose they have no way of contacting the people that bought the AppleCare-in-a-box products and didn't activate.