I think it’s also important not to overindex on universal metaphor.
he, the prince, just found out his father, the king, was killed by his uncle to take both his mother and the crown and he feels he has to face the entire political and royal machine of the country against him - and wonders if it might not be better to just die.
Considering the character is also named after Shakespeare’s son who died (Hamnet Shakespeare, with an n) - it could also be a maudlin / macabre reflection on his own role as a father.