Place of birth in DS2019 and Passport is different. City name A is on the DS2019 and village name B is on the passport.
Is it required to have the same place of birth on DS2019 and passport?
I do not think it is a problem. In a lot of documents (and different countries) we have such cases. The real place of birth doesn't change, but documents often list the official name (as at the time of release of the new documents), so places change.
I assume that either the small village is no longer an official political entity (or just no longer an entity for people registering).
If the village is in the same zone of the city: there should be no problem. (exception on contended [maybe just on the past] territories, especially near boundaries).
BTW: I'm in a western country where there is not the concept of "place of birth" (so it is not listed in any document, official or not). Never had problem with visa and immigration on other countries.