OK, Mulan's whole name in Chinese is Fa Mulan. You pronounce 'Fa' when you are speaking Cantonese, but you pronounce it 'Huah' when you are speaking Mandarin. Both mean 'flower'. Disney did not make a mistake. They just picked a dialect to say the name in, since they could not use both dialects at the same time.
Actually, the story of MuLan is not that different from Disney's version. MuLan left her big sister and her littler brother (hence the dog's name) and her parents to join the army so her father would not die during battle. She was gone for 20 yrs and came back home from battle. Her comrades in the army saw her and then realized she was a woman and were very surprised. The moral of the legend of MuLan (which is believed to be a true story) was, "If you set two rabbits, (one male, on female) on the ground to jump and flee...who would be so wise as to tell if the rabbit were a he or a she?"