Yes, someone has been born a king.
Specifically, John I of France was born King of France. He was a posthumous son, meaning he was born after his father, King Louis X, had died. Because his father was king at the time of his death, John I became king from the moment of his birth on November 15, 1316. Sadly, he lived only five days and died on November 20, 1316.
Due to his extremely short reign, he is often referred to as John I "the Posthumous". While unusual, his situation demonstrates that it is indeed possible for someone to be born into the position of king.
