In current English usage, post-1540s, a minx is not any sort of animal at all. It is an insulting word for a woman who is considered too flirtatious and rude. There is an obsolete English word, mynx, that dates from the 16th century that refers to a playful pet dog.
There is another, and current, English word that is spelled and pronounced similarly to minx that does refer to an animal: mink. Minks are members of the weasel family. There are several species, one native to North America and another to Europe. They are trapped or farmed for their luxurious fur.
The two words, mink and minx, are probably not related in origin. The etymology of minx is well-attested to as being from mynx, for a dog. Mink is of dubious origin, most likely derived from Scandinavian. Dutch and Swedish have similar words for the same animal.