However most ppl incorrectly assume that because it's a social construct then you can identify with another gender and become a real woman or a real man. A chimp can wear clothes, get a job, be treated like a human etc but that doesn't make him human.

Personhood is a social construct, so depending on how a society constructs personhood this chimp could do those things and then become a person.

Gender is a social construct because biology doesn't tell us how we should classify people into gender roles, just like it doesn't give us exact guidelines on what exactly a person is.

Your gender simply isn't a universal fact, just like how "adult", "white", "person" and "moral" aren't universal facts either.

These classifications depend on the cultural context, therefore they are social constructs.

The only way someone can actually change their sex is through an operation and even then the technology is nowhere near close to there yet.

It's transgender, because they are fully aware that they aren't changing their sex.

But they are changing how society perceives them. As long as people don't know their chromosomes and birth sex they treat them according their preferred gender identity.

Sex refers to your chromones. Male and female refer to your sex. Your sex is a biological fact and won't change no matter where you go.

But gender refers to how society classifies you. Man and women (and two-spirit and any other gender that exists in cultures across the world) refer to your gender. Your gender can change if you simply move to a different location, because how society classifies you is just a cultural opinion.

Their are physiological and neurological differences between the sexes that aren't changed by the current surgery or hormones. Basically you would have to have a new body grown for you then operate on the brain to account for neurological differences and finally transfer the brain into the new body. That's going to be a long time from now.

But don't you think that sometimes something can go wrong during pregnancy? Like the wrong sexual hormones, or the wrong amount being present during early development or also bad receptors for those hormones?

Because transgender people evidently have a neurological makeup that aligns with what they claim to feel.

http://sitn.hms.harvard.edu/flash/2016/gender-lines-science-transgender-identity/

Figure 2: Transgender women tend to have brain structures that resemble cisgender women, rather than cisgender men. Two sexually dimorphic (differing between men and women) areas of the brain are often compared between men and women. The bed nucleus of the stria terminalus (BSTc) and sexually dimorphic nucleus of transgender women are more similar to those of cisgender woman than to those of cisgender men, suggesting that the general brain structure of these women is in keeping with their gender identity.

In 1995 and 2000, two independent teams of researchers decided to examine a region of the brain called the bed nucleus of the stria terminalis (BSTc) in trans- and cisgender men and women (Figure 2). The BSTc functions in anxiety, but is, on average, twice as large and twice as densely populated with cells in men compared to women. This sexual dimorphism is pretty robust, and though scientists don’t know why it exists, it appears to be a good marker of a “male” vs. “female” brain. Thus, these two studies sought to examine the brains of transgender individuals to figure out if their brains better resembled their assigned or chosen sex.

Interestingly, both teams discovered that male-to-female transgender women had a BSTc more closely resembling that of cisgender women than men in both size and cell density, and that female-to-male transgender men had BSTcs resembling cisgender men. These differences remained even after the scientists took into account the fact that many transgender men and women in their study were taking estrogen and testosterone during their transition by including cisgender men and women who were also on hormones not corresponding to their assigned biological sex (for a variety of medical reasons). These findings have since been confirmed and corroborated in other studies and other regions of the brain, including a region of the brain called the sexually dimorphic nucleus (Figure 2) that is believed to affect sexual behavior in animals.

It has been conclusively shown that hormone treatment can vastly affect the structure and composition of the brain; thus, several teams sought to characterize the brains of transgender men and women who had not yet undergone hormone treatment. Several studies confirmed previous findings, showing once more that transgender people appear to be born with brains more similar to gender with which they identify, rather than the one to which they were assigned.

Now the thing is that gender is a social construct. We in the western world have a binary gender system because we thought that the Christian God is infallible and thus anything that doesn't fit into two boxes is unnatural and wrong. But other cultures just accepted that biology is more complex and used gender systems that were more flexible and even gender systems with more than two genders.

Biology doesn't tell us that we have to address everyone who has XX chromosomes as woman and everyone who has XY as man. And biology doesn't tell us that intersex people have to be classified as one of both sex either. It's our western culture and Christian beliefs that tells us that we have to do it this binary and inflexible way, but neither biology nor science tell us that we have to continue to adhere to this gender system.

So even though they can't change their chromosomes it's still certainly possible that they can change if society recognizes them as being a man or woman.