We learned that her mother was a queen, but this doesn't matter since she was adopted. And she also was completely unaware of her lineage before the revelation preceding the Battle of Endor.
So, why is she considered a princess?
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Technically, I don't think she could be considered a princess by birth, since her mother Padmé Amidala was no longer Queen of Naboo at the time of Leia's birth, having taken the position of Senator of Naboo. So it's only via adoption that she has royal status.
Leia was adopted by the Royal Family of the planet, Alderaan. Her adopted mother was Queen Breha Organa of Alderaan and her adopted father was the Viceroy (and Senator) Bail Organa. Leia became a princess because she was adopted into royalty.
After the destruction of Alderaan, Leia is still considered their Princess (and assumed Queen) by the remaining Alderaanians, even though she calls herself either Senator or General.
In the Star Wars canon book Bloodline by Claudia Gray, it goes into a bit more depth about Leia's royal roots. As to whether Leia (and her brother Luke) are royal by birth, the novel makes an attempt to answer this question. Here is one quote from the book that I found interesting:
'Leia, I always told you the truth about your mother and how she died. But I never told you that she was Padme Amidala, former Queen and Senator of the planet Naboo.' A war orphan and yet a royal by birth? Surprised as Lady Carise was, she decided it made sense. Leia's nobility was indeed in her blood....
Bloodline by Claudia Gray, page 202
Actually even exhiled Royalty is still consider King and Queen. Like the ones from Greece. Speaking of Greece Prince Philip who you claim was not a Prince by birth was actually born as grandson of both a King and Queen of Denmark and a King and Queen of Greece. As such he was born a Royal Prince. Upon his marriage to the future(Now Queen) Elizabeth he was made the Duke of Edinburgh. Hence Leia is the Princess of Alderaan. Technically is Queen as they have a population that survived the death star blowing up Alderaan.
She was adopted but no one knew this. They believed she was the daughter of a queen by birth. And that isn't changed even though she has no kingdom anymore. When kings and queens are removed from control and a monarchy dissolved they still keep the title and everyone still calls them prince, princess, king, queen, etc.