No, it is totally two different things.
Gender is determined by biology. DNA in particular, but not entirely.
Gender roles are social and determined by environment, society, and the like.
Thus, someone who say cross dresses and acts like the opposite gender is taking on a different gender role while still of their original gender. Someone who goes further with genital mutilation and cosmetic surgery takes that same approach to a new level but remains their original gender. The DNA and biology doesn't change, just the appearance.
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