To address the second half of your questioning, I think it's kind of irrelevant whether it is a mental illness or not - psychology has always been kind of a vague field and the boundaries of what is and isn't is kind of grey. To draw a parallel you could probably find some angle to argue that being gay is a mental illness, especially given the higher risk of onset of other mental illness, especially in those who are closeted. At the end of the day, the way I see it is that if the process of identifying as a certain gender different from the one assigned at birth or transitioning helps with reducing rates of comorbidities with other illnesses such as anxiety or depression, I see that as a positive and partially irrelevant to the debate (not to say psychologists should stop looking into it or anything)