I think it's a bit of a cultural throwback.
As someone pointed out in the original post, there are two "sexes," and they are determined by the absence or presence of a y chromosome. However, gender is more complicated than that. Most of the societies on Earth recognize more than two genders (the "hijras" of India, or the "two-spirits" of some Native American tribes, for example), but as they have become "First-World" societies, the practice of that recognition has faded (even Middle English and ancient Chinese pronouns used to be gender neutral, but that has died out). I think we're subconciously associating multiple genders or non-clearly-defined genders with being "uncivilized."
That being said, there is a distinct difference between the male and female sexes, and biology and psychology are teaching us more every day.
I'll do the best I can with my kids. My daughters like to dress up in girly stuff. They also fight over Matchbox cars, and own their own fishing poles and baseball gloves. I will buy them high-heeled shoes. I will sign them up for ballet, if they want to. I will also not allow either one of them to drive a car until they can check the fluid levels and change a tire BY THEMSELVES, just like my daddy did with me.