Can a person change their gender?
You're mixing up sex and gender here.
Sex is the biological basis of your body. Female mammals have large gametes; malemamals have small gametes. Even people with differences of sexual development (DSDs - often called 'intersex') have this basic biology.
Gender is the set of social conventions, roles and stereotypes of character & behaviour which human society tends to expect of the different sexes. Used to be called "sex stereotypes".
It's important we can make this distinction, because it's what enables feminists to challenge sex stereotypes for women, and separate out our biological potentials (mostly the different roles of females & males in the reproduction of our species) from the expectations & socialisation into how we're told we should behave, according to our sex.
So feminists can say, "Just because women have babies (sex), doesn't mean we are limited to the role of mother and nurturer (sex stereotype)."
Gender is the social system of conditioning predicated on sex. WE need to separate them out.
So yes, I suppose people can change their gender roles, or challenge them: most people do!
But we can't change sex. That is magical thinking.
And the problem with a lot of the gender ideology extremism we're seeing today is that the idea of "living as a woman" is a very superficial and often very offensive & demeaning view of what it is to be a woman.
There's a really interesting bit of the history of science embedded in this: ideas about transition were first made popular at a time of sexism and backlash against women's liberation ...