"A dog acting like a cat is still a dog" is just pointing out how we use the word dog. It says nothing about laws of nature or metaphysical truth of animals. In the ordinary "language-game" we are using right now, we don't classify animals by their behavior alone but by lineage, biology, appearance, etc
Our concepts of 'man' and 'woman' are not like the concepts of 'round' and 'flat' because those are used in a completely different scientific "language-game", settled by measurement, while man and woman function more like family resemblance terms that can shift based on context, ie biology, social roles, identity etc. The meaning of 'man' or 'woman' is not fixed by chromosomes like the shape of the earth is fixed by physics; it is fixed by the language-games we play, and those games can change.
The temptation to link biology, medicine, psychology, etc together to find some single, hidden essence that ties them all together is mistaking how language works. if you look at how words are used in these practices you will find overlapping but non identical criteria, not one essence.
source: wittgenstein