I'm not sure how commonly available a chromosome test is for the average person, so it would possibly be difficult to get that confirmation. Perhaps someone will be along to advise.
It's entirely possible that her chromosomes are not XY, or XX.
You do realise they teach a very simplified version narrative re chromosomes in high schools? An estimated 1.7% globally have different sex chromosomes to the 2 aforementioned categories of xx and xy. In the UK that's over a million people? This whole 'sex is binary' thing is factually inaccurate. We can look down an electron microscope and see that it is inaccurate.
All that being said, you're perfectly right, no amount of surgery will change your chromosomes. In that case, if you're not strictly male or female, what do you do? This is a serious question, I'm actually asking? You've been raised as a boy, you desperately feel that you're a girl, turns out your chromosomes are neither. How are you supposed to participate in society where people literally take to the Internet to discuss how your existence is impossible because their high school science teacher told them so 50 years ago? Please someone explain?