The prince of the kingdom of Persia is most likely Cyrus the great, who started ruling in 559BC. The “kings of Persia” is probably referring to Cyrus and his father Cambyses, who died in 551BC, co-ruling with Cyrus for about 8 years.
It is not uncommon in Scripture for kings to be referred to as princes. Eg – 2 Sam 6:21
And David said to Michal, “It was before the LORD, who chose me above your father and above all his house, to appoint me as prince over Israel, the people of the LORD–and I will make merry before the LORD.