She isn't really a princesses.

Her father was a war veteran and she married a war general, I still love her, but she isn't a Princess

Not royalty and actually fits the heroic paradigma: she's a normal person with great qualities, she wins at fights (against Shan Yu, and Shang at the end of the training). She doesn't have an unusual beginning (like Aurora, for example, or Rapunzel), but then again, we don't know much about her birth and infance.

I definitely think of her as both. She may not be of royal blood or married into it, but being a princess is about more than that. She is considered to be more important than three mere princesses. (Just my opinion though.)

I like her, but she doesn't deserve to be in the Princesses lineup. She's a great heroine, but she has nothing to do with royalty.