She's right to compare them. Both are assigned to you based on some physical and social characteristics and can predetermine your life to an extreme degree.
Ethnicity and gender are very similar in their core, in the way they exist as a social construct, or "collective hallucination" as I like to call it.
What's she's wrong at is thinking either is an immutable biological fact.
Gender is not assigned to you, except in the case of infants or other people who lack capacity to make their own decisions. Our gender identity is something we must each individually discover for ourselves, and it is not constant.
I agree she's wrong about a lot of things though.