Are these correct?
I was born with the wrong gender.
I was born into the wrong gender.
I was born the wrong gender.
Number 1 is grammatically correct. One is born with certain physical attributes, whether a hair lip, a deformed foot or large ears.
Whether you are born with a gender takes you into a different semantic field. Many people argue that while you are born with certain (male/female) sexual characteristics, you later grow into or adopt a gender. They would challenge the notion that you are born with a gender or with the wrong gender.
It is not grammatically correct to say that you were born into a gender or born a gender.
https://www.who.int/genomics/gender/en/index1.html
https://amaze.org/video/gender-identity-sex-at-birth/