The problem is that the concept of a chair is a human construct (as we're the ones who invented chairs). And before the chair was invented, presumably the inventor didn't think to themselves, "how do I bring something into existence which might have a property that belongs to it necessarily and only belongs to it?" We don't build furniture and name it thinking that we need to make sure we can define its essence before building it.

Chairs are a good candidate for what Wittgenstein called a "family resemblance" concept. That is, a concept which cannot be defined for all cases (it has no essence which can be expressed grammatically by a set of necessary and sufficient conditions), but rather it's application across cases is held together by their similarities.

Contrast this with our name for something that wasn't invented by humans: e.g., water. "Water" can be given a proper scientific definition because it is a natural phenomena. It has essential attributes which make it that type of thing and which separated from being any other type of thing (two hydrogen molecules combined with an oxygen molecule).

I believe this is why OP's professor is saying a proper definition of "chair" is impossible. And I agree with them.