
I think the current king isn't of royal blood, he usurped the throne and probably neither his son the second prince and the mc princess are blood related.
The second prince must know and he hates her because even if she isn't the king daughter she is royalty because her mother is a prince's
I said all this because if this if she isn't somehow unrelated to the first prince then they would be cousins in first grade

As of now what we know is that:
Zigrid the second prince and Arianne the MC are half siblings, same father, diferent mothers, but it's implied that maybe they are not as the Zigrid in the new timeline was surprised when Arianne told him that blood siblings can't marry.
Arnold, the first prince is supossed to be cousins with the MC as his father was the king and the blood brother of the current king who is Zigrid father and supposedely Mc father too.
Arnold's father was the king, he died in an accident, I suspect he was killed, and his brother is now the king.
Marianne, Mc's mother, was a princess in the empire who was defeted by the kingdom.
I'm going to need a neat little chart showing how each character is related to each other - i don't wanna accidently ship relatives