Romans 13 enjoins obedience to legitimate civil authorities regardless of the form of government and does not specifically mention kingship (the Roman Empire technically did not have a king per se in the time of Paul). The “divine right of kings” is a theory used to justify the kind of absolute monarchy that arose in the early modern period of European history as feudalism was replaced by the state.
I see that we’re all mostly just talking off the top of our heads, so I shall do likewise.
The notion of a king differs from that of a warlord or chieftain in that the king has a kind of legal or religious sanction that a mere strongman does not. We can see this in the earliest civilizations of which we have records. The Pharaohs of Egypt held power not only through the military power of their armies and the economic power of their slaves, but through the legitimacy conferred on them by the priesthood.
The Old Testament relates that after being led by “judges” who emerged or were called by God to deal with particular crises after they arrived in the Promised Land, the Israelites asked the last such leader, Samuel to anoint a king like neighboring peoples had. Samuel warned them of the dangers of kings, but the people insisted, so Samuel anointed Saul the first king of Israel.
Judging from linguistic evidence, Indo-European peoples seem to have had a notion of kingship before they invaded the Mediterranean world and continued into Europe. I don’t know how much their concepts of kingship were influenced by earlier civilizations. They established petty kingdoms, tribal chiefdoms, and city states in the Bronze Age and early Iron age before the rise of Persian Empire with their despotic rulers. The Persians fell before Alexander, who failed to establish a legitimate line of succession.
Rome had kings in its early days, but the Romans ditched kingship and established the Republic, which eventually became the Empire, which was eventually overrun by Germanic barbarians, who retained those earlier Indo-European notions of kingship. That brings us to the Anglo-Saxon invaders of Britain, who established more of those petty kingdoms that would eventually be united into a single kingdom of England, which the aforementioned William conquered in 1066, but not before taking steps to legitimate his claim to the throne.
The direct lineal descendants of William the Conqueror lasted less than a century before dying out. You can use Wikipedia to learn who has held the crown of England (and later the crown of the UK) since then and how they got it.