"Соседство" is rarely used on its own. You will normally find it in set phrases like "Жить по соседству" ("to live nearby/next door") or "магазин расположен по сосеству с нашим домом" ("the store is next door to our house"), and it is never used in the sense of "a district/area of a city." It simply doesn't have that meaning.
Outside of the above phrases, it can be used to mean something like "nearness, proximity", or "state or condition of being a next-door neighbour".

And on a more general note, don't trust bilingual dictionaries, especially smaller varieties. They usually give some possible translations of a word instead of explaining its meaning(s).