The heart of the matter is, what is gender? It's clearly not the case that gender is identical to being biologically male or female, eg, based on which sex organs are present, or on the chromosome. You need only read the Wikipedia pages on "intersex" and "third gender" to see some of the diversity in the world. Sex indeterminacy is only one of many possibilities.
Go back to the original Turing Test. In "The Imitation Game", the questioner must determine which of two people is male and female based only on written notes. (Presumably typewritten or some other means which doesn't even reveal handwriting.) If the only way to tell that a person who looks like a female is actually a male is through a close physical examination, then will you at least agree that they are doing a good job of acting as a female?
Of course there are many people who do that, including actors and cross-dressers, who see it as a different persona which they can put on or take off. For those who work hard at it, it's a sign of respect to be called "she" because it's a recognition of the effort needed to get the body language, and voice patterns, and dress style down.
But some people feel that being born male was a mistake, a birth defect. A cleft palate is a birth defect which is easily fixed nowadays. Nose jobs and breast implants and LASIK are usually voluntary procedures to change a genetic characteristic. Gender reassignment isn't as simple, but much easier pre-puberty. One worry is that the person, decades later, may decide that the choice made as a child was wrong, or at least deluded. That's why there's a lot of counseling involved.
If someone considers themself a female, takes on female gender roles, and to every extent possible acts like a female, then can you see why there's some offense that some side effects of some stupid little chromosome still make others call them a "he" instead of "she"?
That person may still be deluded. The question for you is, how do you tell the difference between a delusion and (what you consider to be the impossible case of) actually being born as the wrong sex? What are the negatives and benefits of encouraging vs. denying that delusion? Bear in mind that clearly a number of transgender people are happier having made that change.
As to your Superman example, "Superman" is a specific person from a fictional world. But suppose you thought you were Kryptonian, and you underwent hypothetical genetic tinkering and technological augmentation to get x-ray vision, super-strength, and so on. Then yes, I would call you a Kryptonian, or a human transformed into a Kryptonian if I wanted to be more precise. Just like my Dad, born Canadian, is now a US citizen. But if you just decide one day that everyone should call you "Superman", without making any effort at it, then don't be surprised if people don't agree with you. What would that effort look like? I saw a Superman impersonator on the Strip in Vegas, and would have no problems calling that person the Man of Steel.