He would be correct if genders have not been socially constructed. You can say genetically, people cannot change.
But the reality is that the society does subscribe certain qualifies to the two genders, from colour to fashion to interests and profession and everything else.
Personally, I think we should go the other way and stop caring about genders altogether. But as it stands, identifying as a woman and as an elephant are very different. For all intents and purposes, you are a woman if you have the parts and do what the society thinks are women's stuff, but it will be very difficult for Morgan to physically resemble and elephant, let alone behave like one.
Also, if you identify as a non-human animal or an object, of course, people are not going to accommodate to your needs. If you are not human, you have no human rights. If you are not human, you have no civil rights. But of course that is neither here nor there.