She's a Princess because there was nothing that contradicted that back in 1977. George Lucas just happens to be the right kind of stupid to contradict his own fictional works over 20 years later despite the fact that with fiction you just decide what happens and thus can make anything fit.
In Return of the Jedi Luke asks Leia if she remembers her real mother and she DOES despite her real mother dying ten seconds after her birth in Episode III and her "knowing" her real mother just as long as Luke does. That sure as hell doesn't make any fucking sense but, again, "Lucas is a dumbass" is as good an explanation as any. Leia's title makes tons of sense in comparison to something like that.
So I'm guessing the "prequels were a mistake, it's nothing but trash".
This was explained to me: Leia has a different type of Force Sensory than Luke, that's why she could sense his danger on Bespin in Empire Strikes Back. She only becomes "aware" of it after she learns her father and brother were both Jedi (i.e.: knowing Luke was safe in Return of the Jedi). It was told that she can detect the feelings of a person better than most cause of her powers but didn't hone them till much later...something I spect we see in The Force Awakens!
I have heard of that but it is very elaborate explanation for something that in 1983 was easily summed up with "Luke and Leia were separated at birth. Leia was kept with her mother and thus got to know her before she died when Leia was young enough that her memories from then are fuzzy." That's the simple way that anyone would have interpreted that scene when Return of the Jedi came out. This "she has a different type of Force powers" explanation is unnecessarily complicated and is only the sort of thing someone would come up with if they messed up and contradicted their own movie and needed to "fix" their mistake with a retcon. Until now I never even saw the Bespin thing as Leia's powers but Luke using the Force to call out to a friend for help. Who else was he going to call out to? Han's in carbonite, he doesn't know who Lando is at that point, presumably can't use the Force to call out to droids. It was Leia or Chewie.
George Lucas has more or less admitted that he made this stuff up as he went and a lot of writers do that so it's not really anything to hold against him. So Leia is a Princess because that's what he decided when he made Star Wars and it made perfect sense until he tried to fill in the blanks with the prequels. It was his responsibility to make sure that everything fits without resorting to complicated and confusing explanations and he did not do a good job of that. So you can go with whatever explanation he has used or pretend it's 20 years ago when you didn't know much about Leia's parents so you just assumed she was the daughter of a king of some sort.
If we want to get into geeky stuff you figure the title wouldn't come from Queen Amidala because then wouldn't it be common knowledge that she was Anakin Skywalker's kid? Clearly Darth Vader has no idea he has a daughter.