ZIP codes are the route a mail truck takes to deliver mail. That mail truck begins and ends it's route at a specific Post Office. The +4 portion of the ZIP code denotes where along the route something is such that mail can be sorted in the order it will be encountered along the route.

So a ZIP code is an area. A ZIP code is often used incorrectly to apply other demographic information such as race or income, those are generalizations and not necessarily 100% accurate.

The route represented by a ZIP code may enclose an area but the ZIP code itself is not an area. It is a set of delivery points.