She’s a princess by both birth and adoption. And before the destruction of Alderaan her adoptive parents made it clear that she was their one and only heir, and the people of Alderaan recognized her as rightful heir to the crown. Although by the time of the Force Awakens (around 35 years after the destruction of Alderaan) she no longer uses her inherited title, preferring to be addressed by the title she’s earned, general.
And yeah, Luke would be a prince, technically speaking. But it doesn’t appear that he ever cared about such titles.