If you get close enough, it usually gets handled in the local sort, but not always.

On cities, the mailing address city really is the name of the post office that handles your delivery route. Often there's a relationship with the city you live in, but there's cases both ways --- I used to live outside city limits, we had a census designated place name, a municipal sanitary district and had a fire department at one time... but never a post office, so our mailing address used the nearby city name, where our post office resided. The place name had an incorporated city on the other side of the state, so using that wouldn't be great.

Nowadays, post offices often have a list of alternative place names, so where I live now, I can pick between the incorporated city name, the nearby large city where a post office that processes all my mail is located, or any of the numerous small post offices that once served my city.

Bigger cities can have multiple post offices and zip codes with the same mail address city.