I've never run across anyone in the U.S. who refers to postal codes as zip codes.
Say what? Where in the U.S. do you live? I've been all over, and I've never once heard them called 'postal codes'; always and every time 'zip codes.'
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.