The question you are asking stretches into deep prehistory and is essentially an anthropological one, on the origins of social hierarchy. To illustrate, consider one of the more obvious cognates between Latin and Sanskrit - "Rex" and "Raja", from a reconstructed Proto-Indo-European root *reeg-, "one who leads", derived from *reg, "to move straight, lead". These notions go so far back that it is impossible to say who was "the first king". But we can make some observations and try to better understand the origins of social hierarchy in at least this one particular tradition.

The Proto-Indo-Europeans dwelled in the Pontic Steppes, today Ukraine, Russia, west Kazakhstan, about 5000 years ago, give or take. They were likely early domesticators of the horse, and they spread across Eurasia with the proliferation of chariot warfare. We know from their reconstructed mythology and linguistics that their society was tribal and patriarchal.

The oldest Indo-European narrative sources are the Indo-Iranian traditions of the Avesta and Rigveda. They are deeply concerned with political-martial power (khshathra or ksatra) and law/righteousness/the natural order of things (rta or asha). One of the words used to mean "lord" is "asura" or "ahura", usually thought to derive from a PIE verb meaning "to beget", again suggesting a patriarchal organization - the leader of a tribe would probably be the man who is the patrilineal ancestor of the most members. We can infer something similar from Greek mythology, where the titan Cronus castrates his father Ouranous in overthrowing him, robbing him of his generative power and thus his fitness to rule as patriarch.

Anyway, there's a great hymn of the Rgveda (4.42) which, once the core tradition is extracted, shows some of the things on the Indic peoples' minds. It can be reconstructed as an exchange between the Asura Varuna (god of the oath) and the Deva Indra (a great warrior and the most prolific deity in the Rgveda, with a position similar to Zeus):

####Varuna:

1. Lordship (raṣtṛam) belongs indeed to me, the eternal sovereign (kṣatriyasya), as all the Amrta acknowledge to us//The Deva obey and follow the will of Varuna, I am the king (raja) of men's most lofty refuge

2. I am King (raja) Varuna, I was granted these primordial celestial powers (asuryani)//The Deva obey and follow the will of Varuna, I am the king (raja) of men's most lofty refuge

4. I let the dripping waters rise up, through rta I uphold the sky//Through rta I am the lord who rules through rta.

####Indra:

5. Men who drive swiftly, having good horses, call on me when surrounded in battle// I provoke strife, I the mighty Indra - I whirl up the dust, my strength is overwhelming

6. Such are my deeds. No divine (daivyam) power can control me, the invincible one//When draughts of soma, when songs have made me drunk, then both the unbounded regions frighten.

Here Varuna - often taken to have been a high god of an old Indo-Iranian pantheon, possibly in conjunction with Mitra, god of the covenant - explains that by the natural order of things (rta), rulership is his. Indra on the other hand, suggests that he is too powerful to be controlled by such concerns. Indeed, this is one of the basic points where Zoroastrianism breaks off with Vedic tradition, in so to speak rejecting Indra's claim to rule and more generally rejecting martial virtue. One cannot of course reconstruct history from mythology, but the notion of military prestige upsetting the social order is very old indeed, and in their encounters with settled societies like the Indus Valley Civilization, it is easy to see how the idea of a young conqueror upsetting social hierarchies could be a social reality for the ancient Indic population.

So, to wrap up, in this particular tradition, the Indo-European one, the notion of kingship seems to trace back to the patriarchal organizations of clans or tribes, where social order would be maintained by deference to the most powerful male in the extended family. This is of course not universal, but patriarchal notions such as "the father is the master of the house" are, I would argue, highly cross-cultural.

An excellent book if the Proto-Indo-Europeans piqued your interest is The Horse, the Wheel, and Language.

Let me know if you have any followups or if you feel that there's some aspect of your question that I could address better!