I think the general consensus over the years has been that Bail Organa held a royal title, and Leia warranted the title of princess by virtue of pedigree. But that's not always the way it works. She might've been princess by virtue of marriage to a prince, or by popular acclamation/election like her mother. Or Alderaanian royal titles could be accorded on the basis of democratically-elected or appointed positions -- planetary heads of government are acclaimed kings, representatives to the Imperial Senate are acclaimed prince/princess and granted ambassadorial rank, lesser officials are created archdukes, marquesses, etc. In this way, a modern society could maintain the grandeur and tradition of royalty, without the bother of a royal family.
Interestingly, See Threepio, a protocol droid with years of experience among high-ranking Alderaanians, calls Leia, "Your royal highness," in Return of the Jedi. The style "your royal highness" is used for non-reigning royalty; a sovereign is referred to as "your majesty." This is in accord with Leia remaining a Princess and not ascending to Queen of Alderaan (or perhaps Queen of the Alderaanian People, etc). From this we may conclude either that Leia has some relative who ascended to the throne after Bail Organa's death; royal titles are not based solely on pedigree but on some other source of authority or status; or Leia has declined to become the sovereign of an essentially nonexistent state.