Mainly, the issue is that people think they're required to give out their ZIP codes when asked because it's needed for their credit card to go through. In fact, it's just used for marketing and helps connect your name with your address, in ways that aren't quite clear to mortal humans, but the credit services understand.
For instance, he quotes the Fair Isaac Corp., which deals iwth FICO credit scores, and offers a service they claim can increase direct market success by up to 400%. “FICO Contact Builder helps you overcome the common challenges of gathering contact information from shoppers—such as complicating or jeopardizing the sales process by asking for an address or phone number, or complying with regulations. It requires minimal customer information captured at point-of-sale, just customer name or telephone number and the customer or store ZIP code.”
In other words, customers may not want to give out their address or home phone number, but will readily tell you their ZIP code, and with that the store can get the same personal information it needs.
The article has an amusing story (amusing if you weren't participating in it) about a woman who didn't want to give out her ZIP code to a store that really, really wanted it. As Tanner notes, at least in California (he doesn't make it clear about other states), the California Supreme Court ruled in 2011 that stores cannot demand customers give their ZIP code. Of course, if a store in your state insists on getting your ZIP code, and you can't refuse, what you can do is walk out without purchasing the product...