It's worth noting that the story heavily implies that Amira is lesbian. It's absolutely explicit that she finds the idea of a (heterosexual) marriage "monstrous".
Note how her previous ideas, before "chose me a prince," let her avoid marriage entirely. And even "find me a prince" could be a purely political marriage. So really any conclusion forcing Amira into any marriage, is pretty much "unspeakable."
While Kitkat's answer is excellent, and the threat of incest may well be implied here, I think that option dulls the very open criticism towards power dynamics we consider far less shocking. The keys to the portrayal of Amira's story, in my eyes, are:
Once upon a time there was a rich king who had no sons, and whose only daughter was too beautiful. She was so beautiful that men could not stop themselves from reaching out to touch her in corridors or following her to her rooms, so beautiful that words of desire tumbled from men’s lips like diamonds and toads, irresistible and unstoppable. The king took pity on these men and drew his daughter aside, saying, Daughter, only a husband can break the spell over these men; only a husband can prevent them from behaving so gallantly toward you.
When the king's daughter is threatened with rape and harassment, the king takes pity on the men, and lays full responsibility on the shoulders of the daughter.
And this simple aside:
The king’s daughter, who did not want a husband
A categorical statement. Not "didn't wish to marry against her will," but rather "did not want a husband," any husband.
The horror of this sub-story is the way harassment and rape are considered "inevitable," and "the woman's fault," that she is held responsible for "fixing" the situation regardless of the cost to her. (And, let's be honest -- marrying, in such a society, would be little protection indeed; maybe looking more proper outwardly, but not actually keeping pawing arms off of her.)
This kind of horror doesn't need incest to make it worse, or anything else that isn't already clearly laid out. The situation itself is the horror; succumbing to it is unspeakable.