Hello, I'm Japanese (Mulan is just a handle name) so I may be able to help you a little. In the most common use, both of 王女 and 姫 means "princess". But there's a slight of difference. 王女 means "daughter of king", but 姫 means "daughter of somebody noble". Somebody can be Shogun or Lord or else. So, princess Anne from Frozen is 王女 and 姫 at the same time.