I think you'd have to define what it means for a category to be valid. The way I see it is that a category is either useful or not and whether it is or not depends on our goals which depends on what we like and want which is subjective and the judgement itself relative to such a system so designed. That's not to say they are arbitrary, because they're objectively useful or not with respect to the goal.

I don't think biologists would agree with you about the horse. It's actually consider a crisis in the philosophy of biology that the concept of species is ill-defined. [1] Like I said, the resolution to me is pragmatic: the definition will depend on whether it is effective relative to some purpose.

1. https://plato.stanford.edu/entries/species/