Zip codes relate only to the mail system and have absolutely nothing to do with political boundaries. Some cities will be covered by more than one zip code; some zip codes will cover more than one city.
Some towns in NC cross county boundaries. For example, Cary crosses into Chatham county (from Wake county), and Chapel Hill crosses into Chatham and Durham counties (from Orange county).
Also note that mailing addresses may not accurately reflect the actual town a property is in.
Bottom line - don't use mailing information (city name or zip code) to determine what jurisdiction a property is actually in.
Property tax rates may vary within a city if, for example, there are fire district rates or school district rates. In Chapel Hill and Carrboro, there are school district rates for those portions of Chapel Hill within Orange county. See this table for more confusing data:
Orange County NC Revenue Department