This picture is nice as far as the gender/identity stuff goes, but why is it trying to represent biological sex as a spectrum? And what in the hell does "Asex" mean? Biologically speaking, you are either male or female. Even the various developmental abnormalities and hermaphroditisms are scientifically defined and understood. It's not some nebulous scale of infinite possibilities like they're describing.
Asex would mean "asexual", and as BeesEight pointed out there are a variety of ways of defining physical sex. And while they are certainly uncommon (and in terms of their gender presentation, most of them simply present as whatever gender they identify as), but people whose bodies present an unclear picture of their sexuality isn't unheard of. You're right that the various developmental abnormalities and intersex conditions are defined and understood, but they still represent states of biological sex that are not wholly male or wholly female, but are various degrees of one or the other.
Science hasn't really known the answer to this, but like I said some studies of the brain have shown that a big part of gender identity is based on differences in the human brain. As in, we have male brains and female brains. This partially explains why most boys, for example, are worse at language studies than girls at the same age. Because the areas of the boys' brains that are connected with language develop more slowly. This is also partially why boys are better at math (although a large part of that is also gender stereotypes).
So basically, this kid, who ended up with a male body, most likely has a female brain. Which makes her a girl. That's where the science is heading, at the moment. And since they can't fix her brain, the easiest thing to do is to fix her body to match it.
Well, it is true that there are differences that are caused by prenatal sex hormones, but researchers have found (by measuring the testosterone in second trimester amniotic fluid that was obtained during amniocentesis during pregnancy and frozen) that children's whose prenatal hormonal environment had had higher levels of testosterone displayed greater hemisphere specialization and were more strongly right-handed. And boys who had been exposed to higher testosterone levels of the womb had stronger right-hemisphere associations for emotion, which is more often found in women (it is one of those odds quirks that an increased amount of a hormone so strongly associated with men results in a brain pattern that more closely resembles women).
And interestingly regarding the bolded part, I'm aware of one study of gender differences in early literacy (kindergarten through fifth grade) on four measures that found that there were no significant differences on three of them, and on oral reading the only year with a statistically significant female advantage, fourth grade, which had disappeared by fifth grade, and they were left with a mean difference of less than one word per minute.
So I wouldn't get too invested in the importance of those
average structural differences:
Regarding differences in ability:
It is true that males widely outnumber females at the genius end of the mathematical spectrum. But does that mean that males are, on average, more mathematically capable and females more verbally capable? Janet Hyde, a psychologist at the University of Wisconsin, has conducted a massive amount of research about over 1.4 million people and included writing, vocabulary, reading comprehension. She found no gender differences in verbal ability. But when she analyzed one-hundred studies of mathematical ability, representing the testing of nearly four million students, she did find some modest gender differences. In the general studies, females outperformed males in mathematics, except in those studies designed only for the most precocious individuals. What Hyde and her colleagues - and virtually every single study ever undertaken - found is that there is a far greater range of differences among males and among females than there is between males and females. That is to say that the variance within the group far outweighs the variance between groups, despite the possible differences between the mean scores of the two groups.
And regarding differences in the brain itself:
But that doesn't stop some popular writers from dramatic and facile extrapolation. Here's Robert Poole, from his popular work Eve's Rib: "Women have better verbal skills than men on average,; the splenium seems to be different in women and men, in shape if not in size; and the size of the splenium is related to verbal ability, at least in women." In fact, there seems to be little consistent evidence for significant brain differences between women and men. Jonathan Beckwith, professor of microbiology and molecular genetics at Harvard Medical School argues that "[e]ven if they found differences, there is absolutely no way at this point that they can make a connection between any differences in brain structure and any particular behavior pattern or any particular aptitude.
If there is no evidence of these arguments, why do they persist? One brain researcher, Marcel Kinsbourne suggests that it is "because the study of sex differences is not like the rest of psychology. Under pressure from the gathering momentum of feminism, and perhaps in backlash to it, many investigators seem determined to discover that men and women 'really' are different. It seems that if sex differences do not exist, then they have to be invented."