While she very likely said this in a mean spirited way, I think it’s a very good analogy.
There is no race on your ID card because it would be stupid (and very dangerous as it can and would be misused) then why is there gender? If a person identifies as “black” (whatever the hell that means, are they African? African American? European? South Indian? Etc. - you can easily be born with dark skin on pretty much any place on earth for whatever reason) then so be it, they may have valid reasons to feel a connection with a particular culture or ancestry. Why should a person not be able to identify as “male” or “female” without it massively affecting their lives?
They are all social constructs anyways, people should be free to identify however they want without needing to declare it to the government, in my opinion.
As a person that also lives in northern Italy, and that have a girlfriend with a series on food intolerances, I can confirm with what OP already said. If you are celiac you can find options now days (obviously much more if you live in a city). If you are lactose intolerant you can also find options (Italy is also a place where lots of cheese is consumed, and many recipes involve some kind of cheese or cream). If you are both at the same time, well that’s where problem starts appearing.
If by any chance you also got a vegan in the group well it’s better to just cook yourself (obviously exaggerating but still your options of places may be very limited).
All this to say: you can find options now (when I was a kid it was unheard of), either with big restaurants/pizzerias that have a double kitchen, or with dedicated places that are fully gluten free.