A number of fantasies set in fairytale land -- at least in my experience -- talk as if all princesses must marry princes, and vice versa. It is a serious constraint on the characters.
But if you fare forth from the keep of modern fairytale fantasies, over the vast kingdoms, on a quest for the source tales, you will soon find that the princes and princesses in those tales is far in excess of the heroes and heroines of royal blood in the source material, not for the lack of royal blood, but for the number of non-royal characters
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For once, Disney is not too much to blame. Cinderella is a merchant's daughter, as she is in the folklore. If some of the variants do have princesses -- after they run away from home and become scullery maids -- others have peasant girls. Except for Tangled, all the other movies merely bring in princesses and princes from the tales, and in Tangled, if they made Rapunzel neither a peasant girl nor an ogress's daughter as in the variants, they did turn the prince into a poor lad.
Many more poor lads and lasses can be found in the folk tradition. The peasant lad in Jack and the Beanstalk, the woodcutter's children in Hansel and Gretel, the miller's daughter of Rumpelstiltskin, and the miller's son of Puss in Boots are joined by soldiers, shepherd boys, peasant girls, merchant's sons and daughters, and more who find themselves on quests for golden birds or captured by ogres, going up the slopes of a glass mountain or hiding the forest with a deer.
As well as princes and princesses, of course. There are many prince heroes, and princess heroines, and they may, in fact, be the largest group of heroes and heroines. They are just not in exclusive position.
Beauty and The Beast is a little odd and therefore useful as an example. In the original, literary version, she's a princess whom a fairy smuggled into a merchant's family to protect her. Along with the other convolutions of the tale, that quickly got pared out while bringing the tale down to a less literary and more tale version. This is not because princesses can not marry animal bridegrooms. Many a king has promised his daughter, either because the request was riddling, and he did not see through it, or as a consequence of trying to fulfill her request, as the merchant did while picking the rose, but those do not raise her in another family. While there are folk fairy tales of princesses who are raised in ignorance of their birth, that and the animal bridegroom do not fit nicely together.
The original tale having Beauty be a secret princess is not, in fact, unusual with the précieuse fairy tales. Heroes and heroines, for them, are always of royal blood. If the plot required that the king and queen abandon the princesses in the woods out of fear that keeping them would mean they all starved, after the manner of Hansel and Gretel, the author would describe them as a king and queen in exile and poverty. These writers certainly gave the name "fairy tale" to the genre, but also serve as an example why it is wiser to use folk tales for your source for what fairy tales are like.
The Beast's being a prince is preserved because a royal love interest is, indeed, the archetypal fairy-tale love interest. Indeed, just about the only way to escape it is if the love interest is a shape-shifter, or the child of the villain (or servant, or captive, or lives in the villain's household without any real explanation why), or, of course, both at once. On the other hand, neither one is a guarantee that the character will not have royal blood. The evil wizard can also be a king, so that his beautiful daughter who can turn into a swan is all three, or he could have kidnapped a princess as his servant. It's merely possible.
A few other possibilities exist. I note that noble blood (or sometimes just owning the big house and owning broad acres) is a little funny. Just as socially, nobles are higher than commoners, and lower than royalty, noble characters can be either higher than commoners in the tale (and so serve the plot function of princes and princesses) or lower than royalty (and so serve the plot function of commoners). When Catskin's noble father insists on marrying her off to some man to be rid of her, she asks for three marvelous gowns, and then a coat of cat-skin, with which she can run away, become a scullery maid in a noble household, and go to the ball to win the son of the house, just as if he and she were royal instead of noble. When Tattercoats is oppressed in the household of her noble grandfather, she nevertheless gets to the ball and marries the prince, just as if she were a peasant girl, or a merchant's daughter.
But now and again, there is a tale where a commoner character wins the hand of a commoner character. In Bearskin, whatever happens in the variants, the soldier marries the daughter of an old man whose debts he had paid. In a few French tales, such as The Damsel With the Long Nose or The Little Soldier, the princess behaves so badly that the hero goes back and marries a commoner woman who helped him when he met her along the way. They are not quite as easy to find a firebird, but they are out there.
It's amazing what the quest through fairy tales can find.