If this had been a true story, Mulan would not cut her hair, and while the story is fictional, and will certainly contain some fantastical elements, the decision was made to try and give the new remake a bit more of a realistic feel because of the transition from animation to live-action.
Which is not to say the film hasn’t received some criticism for some potential historical inaccuracies.
And while Disney may have done a great job with the scene in order to make it so memorable, the fact is that what it’s not is accurate.