Greene County is just another county in just another US state. In it you find towns, villages and "census designated places". The first line of contact for all kinds of administrative issues is the county.
For a long time, it was the state who was considered the right level of "is in the administrative-territorial entity" in this case it is New York. This has been remedied and at the same time many other villages, towns and whatnot have been associated with the lowest level of an administrative-territorial entity they have to deal with.
When you reason like the Reasonator, you can go up the food chain and find the state, the country a place like Climax has to deal with. It is just one way the Reasonator makes information for you out of the Wikidata data.