You must live in a different U.S. than everyone else. It's clearly the other way around.
https://www.google.com/search?q=site:usps.com+"zip+code" returns 120k results
https://www.google.com/search?q=site:usps.com+"postal+code" returns 372 results
https://www.google.com/search?q=site:usps.com+"zip+code" returns 120k results
https://www.google.com/search?q=site:usps.com+"postal+code" returns 372 results
What's wrong with calling them Zip codes? That's how they're written everywhere.