District is a more general term that can be used in many different places, but borough is only used in certain big cities, as you said. For example, New York City has boroughs like Manhattan, Brooklyn, the Bronx, etc. Even lots of big cities don't use boroughs - I'm from Chicago, and there isn't a borough system there. I would recommend using the terms "neighborhood" or "district" for most places.

Kitty gave you a great answer. The word “borough” is funny in the cartoon precisely because we do not expect Springfield to have boroughs — as if it were comparable to NYC. Also note that in real life, they would simply name the “borough” rather than describing it. It’s a joke. In reference to Warsaw, you can probably use whatever the local term for the local political districts is.