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It is important to differentiate gender from sex. There are only two biological sexes, although humans are known to survive certain aneuploidies of the X and Y chromosomes, producing viable and sometimes fertile individuals with various numbers of complete or incomplete X and/or Y chromosomes.
In addition, there are many known possible intersex conditions caused by problematic cell division, among others; in that case the patient's tissue may contain mixed cells with a disparity of X and Y chromosomes. Like a chimera. But completely different.
Traditionally, surgery was performed at the time of birth to assign a single gender if certain conditions were met.
All that said, when someone identifies as a third gender or bi-gender individual, they're not necessarily referring to a congenital condition (such as hermaphroditism). In many cases, they merely feel that they do not possess a strictly male or female gender role or identity - for example, there are eunuchs or nullos who consider themselves as something other than male and have undergone professional or amateur surgery by choice.
And that's science! Probably. I can't be bothered.