According to Wikipedia, Chenango is an Oneida word meaning "bull thistle." There are several places with this name in North America, including one as far south as Texas.
My limited understanding of the Amerindian linguistics is that the languages are numerous and quite diverse, with even geographically proximate tribes often showing great linguistic distance. This, along with the geographical distance from New York to Mesoamerica, would suggest the -nango similarity is likely coincidental (or a remarkable borrowing). But there is one linguist Greenberg who sorts the Native American languages into three (very broad) macro families. Greenberg's work is not well-regarded by most linguists (since Greenberg never provided detailed reconstructions to account for common origin and divergence), but it makes some sense from an anthropological perspective (i.e., it seems to correspond to the human migrations that have been reconstructed from archaeology, population genetics, etc.)
Most linguists concerned with the native languages of the Americas classify them into 150 to 180 independent language families. Some believe that two language families, Eskimo–Aleut and Na-Dené, were distinct, perhaps the results of later migrations into the New World.
Early on, Greenberg (1957:41, 1960) became convinced that many of the language groups considered unrelated could be classified into larger groupings. In his 1987 book Language in the Americas, while agreeing that the Eskimo–Aleut and Na-Dené groupings as distinct, he proposed that all the other Native American languages belong to a single language macro-family, which he termed Amerind.
Historical linguists also reject the validity of the method of multilateral (or mass) comparison upon which the classification is based. They argue that he has not provided a convincing case that the similarities presented as evidence are due to inheritance from an earlier common ancestor rather than being explained by a combination of errors, accidental similarity, excessive semantic latitude in comparisons, borrowings, onomatopoeia, etc.
However, Harvard geneticist David Reich notes that recent genetic studies have identified patterns that support Greenberg's Amerind classification - the “First American” category. “The cluster of populations that he predicted to be most closely related based on language were in fact verified by the genetic patterns in populations for which data are available.” Although this category of “First American” people actually also interbred and eventually contributed the majority of genes to both the Eskimo-Aleut (60%) and Na-Dené (90%) populations.
One counterintuitive aspect of determining linguistic relationships is that many legitimate relationships look like a real stretch, especially since the common sound shifts like p/b/f or t/d can disguise the similarity. You really have to squint to see some of those Indo-European cognates for isolated word pairs, yet the sheer breadth and volume of the similarities is ultimately convincing. Conversely, you can get words that look really similar in isolation but end up being false positives. Place names seem like a particularly interesting thing to look at since the names often persist even long after a language becomes supplanted.