The USPS refers to US postal codes as "ZIP codes" only because they felt it was necessary to trademark the term "ZIP code" (short for "Zone Improvement Plan"). The trademark has since expired. Now we're stuck with a proprietary term that is of little value since it refers only to a specific subset of all postal codes, when there's nothing special at all about that subset.
I consider it to be better practice to refer to all postal codes as simply "postal codes".