It has nothing to do with giving up cultural identity. Speaking as an immigrant, I gave up my citizenship of birth when I became a US citizen. I still celebrate the holidays, but my life is here now and that's what makes the most sense. It's stupid to be paying out and creating all kinds of tax headaches just simply to hold on a piece of paper. We have to remember that it won't just be her. With shared income between spouses, this could create headache and potential tax liability on Harry as well once they become a married couple. A lot of ex-pats are actually turning in their US passports because of the tax issue. It has nothing to do with her parents moving to UK or not either.
And actually, there has been conflicting reporting all day from the royal correspondence due to their interpretation during the briefing. New York Times reached out to KP and the statement they got is that she will remain US citizen during the process of obtaining her UK citizenship. No comment on whether she will be renouncing her US citizenship once that's done. Several royal correspondence have also clarified later that no decision regarding her US citizenship has been made on she becomes UK citizen.
As I said, keeping her US citizenship after she becomes a British citizen won't be seen positively by some people in the UK and might be used to attack the Royal Family, which would be unfortunate. That is why I think she will be advised to give it up when she can of course (as someone mentioned, she can't be stateless, can she ?).