It is not true that North Korean leaders claim to never defecate. This is a misquotation from the book The Aquariums of Pyongyang, which used exaggerated terms to describe the mindset of North Korean children. The "perfect golf game" is not true either: it was the result of a local reporter not understanding how golf scores are tallied and accidentally recording bogeys as holes-in-one.

I guess the closest thing to what you are thinking of is the imperial cult that arose after Rome began occupying other territories. Greek and Roman monarchs were sometimes treated as agencies of divine authority, with statues and temples being built to honor them. However, recent scholars such as Mary Beard and Robert Parker (On Greek Religion) have argued that this did not imply that the Greeks or Romans believed their kings or emperors to have superhuman abilities, but was simply part of Mediterranean custom of paying respect to powerful forces. The practice of magic was associated in Rome with insanity and barbarism, so it would not have been beneficial to have people see emperors as magicians (see Georgios Andrikopoulos, "Magic and the Roman Emperors").

(edited to add) From the 1300s through the 1800s, there was a popular belief that European kings could heal through their touch, by virtue of their worldly legitimacy. In later centuries, Protestants accused this "royal touch" of smacking of Catholic superstition; it did not fit with their vision of how kingship should work. Blessings for healing were also practiced occasionally in various Catholic kingdoms. My apologies for not remembering this in the original answer.

The idea that the European "divine right of kings" gave them supernatural powers is a misreading of the concept, which is actually an early modern one. In Catholic Europe, the rights of kings were legitimized by the Church, which presided over the spiritual matter of determining God's approval of violent acts of war and so on. Protestantism eliminated this institution, leaving it up for debate why Europeans had kings at all. Protestant monarchists were therefore logically forced to claim that the king did not need the Church because he was legitimated directly by God. This sort of argument began after the Protestant Reformation in the 16th century so it was accompanied by increased rationalization of the king's prerogatives, not a growth in irrational and outrageous claims.

Whether in England, France, Spain, the Italian peninsula or the HRE, kings and princes worked closely with local aristocrats to achieve success and were never towering above in the heavens. There was no situation where it would make sense for the gentry to believe that the king could turn water into wine or play a perfect golf game. Incidentally, this is also the case in modern North Korea; Kim Jong Un does not pretend to be a magical despot, but proclaims that he serves at the pleasure of his people and since Covid-19 began he has been regularly admitting his failures and requesting forgiveness from the public.