I can't imagine too many people actually define their gender based upon their chromosomes. Until recently, who knew? Probably most of the 6+ billion people alive today, let alone the 80+ billion living in the past, couldn't tell you what a chromosome was. And of those who can, how many have actually had theirs examined to verify they are exhibiting the correct gender expression based upon their genes? Or have had parents do so at their births to ensure they were reared so as to conform to the proper gender role? Realistically, people can't and don't define their gender based on knowing about their 46th chromosome, because they don't have that information.
Maybe it makes a bit more sense if we define our gender based on how our genes manifest themselves through our bodies. We can see that, at least when we're naked. Unless the child is one of those 1-in-2000 born with anomalous genitalia (i.e. intersexed), a doctor or midwife feels pretty confident declaring, "It's a girl" or "It's a boy" after a simple external examination.
They are, however, on occasion, wrong. Take, for example, people with CAIS or Complete Androgen Insensitivity Syndrome, one of the many possible variations of the 46th human chromosome. (Androgen insensitivity as a syndrome exists along a spectrum. Those with PAIS or Partial Androgen Insensitivity Syndrome have a range of response, from near normal to just short of complete insensitivity.) These CAIS children look like girls, act like girls (with all the variation that implies) and carry an internal sense that they are girls. Then puberty hits. They develop ample breasts and a womanly figure. But they never begin to menstruate. Concerned, their parents take them to the doctors, who finds these girls have no uterus, perhaps incomplete vaginas, and instead of ovaries, undescended testicles. These girls are genetically male. They have a fully functioning Y chromosome, which triggers the appropriate production of androgens in utero and later in life. However, a defect in the X chromosome prevents the body from responding to these androgens.
In a normal XY fetus, at about 2 months after conception, a "hormonal wash" of androgens (the first of many) begins a masculinization process. If this "wash" does not occur, as in normal XX fetuses, or if the androgens cannot affect it, as in an XY fetus with CAIS, the fetus continues development as a female. The Y chromosome triggers the production of androgens after birth as well, most notably at puberty, but the bodies of these androgen-insensitive XY people cannot react to them. They don't even grow hair in the armpits and pubic area, which is a response in normal XX people to the androgens their bodies produce. However, while those with CAIS are sterile and may need some reconstructive surgery to have comfortable vaginal intercourse, they for the most part look and act like "normal" women. To declare them men based solely on their XY chromosomes would be absurd. They don't look or behave like men, nor do they have that internal sense of being men. As far as they are concerned -- XY chromosomes notwithstanding -- they are women. And this is only one of the many variations of chromosomes and sex in humans.
The drawback of inflexibly linking gender with chromosomes or even genitalia/reproductive organs is that it misses all that spiffy diversity found both biologically and culturally. Because while gender expression might be tied to the physical sex of the body, it is, essentially, a social role and an individual's sense of who he/she/(it?) is. Human beings, as a species, do not have one way of being a man or one way of being a woman. How we play those parts is determined by what culture we're born into. And many cultures allow for more than two genders. They may have the concept of a third sex, such as the hijras in India, the two-spirit people or berdache of the Native Americans, and the gallae of ancient Rome. Some, without the concept of a third sex, have acceptable means for someone to opt out of one gender into another. An example is the sworn virgins of the Balkans, female-bodied people who become, for all intents and purposes (except siring children), men.
Most of these people have (or had) "normal" male or female bodies, however. To those of us raised in a modern Western society, the idea that there are two genders -- man and woman -- based upon physical sexual characteristics -- male and female -- is so obvious that it's tough for us to tease what are actually two separate things apart. It seems so intuitive and, well, common sense. It's in our cultural atmosphere, and we breathe it in like we breathe in air, without giving it much thought. So why would these people think their gender to be anything other than what their bodies signaled to the rest of us?
We don't know. For some, it may simply be a feeling of being psychologically constrained by the roles dictated by the bodies they wear. However, there is some fairly recent research that hints that hormones present in utero influence not only what form our bodies take, but where our sexual attractions will lie and what gender we will perceive ourselves to be. It's possible our sense of our gender is formed, in part, in the womb. Those whose bodies are congruent with their internal sense of gender would have very little cause to question it and the role their culture prescribes they play. Those whose sense of gender and body are incongruent will find themselves out of place in their own skin. Some cultures provide social roles within which these people find relief for this gender dysphoria. Many others do not. Western society, for example, is highly suspicious of those who don't or can't conform to the behavior assigned to them by their biological sex. (Note the mockery and viciousness faced by many homosexuals (who violate the expectations of which sex each gender will be attracted to) and transgenders.)
We carry our sense of gender within us, just as we do all those other traits which make us us. For those who are comfortable, both in their skin and with what their culture expects from them, the conflation of physical sex with gender expression may seem intuitively obvious. That others do struggle with a dissonance between body and gender offers the insight that it's not quite as obvious as we might think, spurring us on to ask questions we never would otherwise.
Biological Exuberance, Bruce Bagemihl, Ph.D
Brain Sex, Ann Moir, Ph.D. & David Jessel
Gender Blending, edited by Bonnie Bollough, Ph.D., R.N., Vern L. Bollough, Ph.D., R.N. & James Elias, Ph.D.