No, she didn't "win" that debate. Did she prove there are more than two genders? No. No she did not. No matter how many different ways people choose to "identify", there are only two genders. That's what his argument is. Not, 'there are two ways to identify.' The ways a person can choose to identify are endless. But with gender there are only two.
Actually, he proved she couldn't prove him wrong. That's all there is to it. They don't like to acknowledge that biologically there are males and there are females. Her argument wasn't better because she has no argument. There are a small percentage where nature mixes it up a little but just like nature only gives some people three fingers on a hand. We don't label them a new species though.