For a pre-scientific bronze age people there wouldn't have been much interest in any kind of rigorous classification. So "kind" doesn't fit into any taxonomical system. But ancient people were perfectly aware of the fact that some things were different from other things. A leopard wasn't a lion, a wolf wasn't a jackal, a falcon wasn't a vulture.
As such "kind" simply means "kind", in a basic, non-technical sense. There are different kinds of things, ancient people could make those kinds of distinctions by simple observation. So when the Bible says "kind" it just means "kind". It's a non-technical, non-scientific, non-taxonomical term to speak of different sorts of things. The point of speaking about "kinds" entering the ark is to speak about how the different sorts of animals were brought on the ark. There's no special, hidden, esoteric meaning behind it. Different kinds means different kinds, on the basis of simple observation of different kinds of things existing.
If you can tell the difference between a an oak tree and a birch tree, then congratulations, you've recognized two different kinds of trees. And you don't need to know their taxonomical rank in order to make that distinction in your observation.
-CryptoLutheran