This is a fictional character in a heavily fictionalised setting.
Homeric epic portrays Greek leaders, basilēes, as having a non-constitutional leadership role, characterised by prestige sustained through personal qualities such as strength and heroism, wealth, and gift-exchange. In classical Greek basileus simply meant ‘king’, but in Homer being a basileus isn’t technically a hereditary position.
https://www.jstor.org/stable/643251
But, because personal qualities are imagined as hereditary, it kind of is hereditary anyway — just not in an official way. There’s a great article by John Halverson, ‘The succession issue in the Odyssey’ (in Greece & Rome 1986), which lays out nice and clearly that there’s nothing hereditary as such, that basileus emphatically does not mean ‘king’ in Homer, and that the suitors wanting to marry Penelope aren’t doing it because that will make them king, they’re doing it in order to stake a claim to Odysseus’ property.
The proper translation of basileus in Homer, Halverson argues, is ‘big man’, a term coined by the anthropologist Marshall Sahlins to refer to the most prestigious person in a Melanesian tribe.
Homeric epic is 7th century BCE poetry, telling a story set in a dim past lost in the mists of antiquity. Homer society is a mash-up of contemporary 7th century BCE Greek society, and a hodge-podge of false archaisms thrown in to give it a flavour of ancientness. Imagine a film with characters using ‘thee’ and ‘thou’ in an American accent and getting the grammar wrong, engaging in mediaeval-style warfare but never using mediaeval tactics, putting woad on warriors’ faces a millennium out of place, scenes in ancient Babylon by the seaside and with mountains on the skyline — in short, think of pretty much any popular film set in antiquity or the mediaeval period. That’s what Homer is like.

By contrast, when Hesiod uses the word basileus (roughly contemporary with the Odyssey), it doesn’t look like he’s talking about ‘big men’, but about something more like a member of a juridical or governing body — a political structure with elements of a senate and a supreme court.

The upshot is that a Homeric basileus is a creative reimagining of an imaginary past. (A Hesiodic basileus is a creative reimagining too, but with a different flavour. Hesiod isn’t telling a story set in the mists of antiquity, he’s caricaturing political figures in the present day.)

And because it’s imaginary, there’s no need to go looking for underlying constitutional principles.

In the Homeric picture, leadership is something you earn by
being the son of a previous leader
being super strong, super good at fighting, super clever, etc.
being the richest person around
and most importantly, by
leading people, and
hobnobbing with other leaders.
Yes, that’s tautologous: you get to be leaderly by leading people. But that’s largely how it works. Telemachos earns a leaderly position in the Odyssey by (a) summoning an assembly (book 2), (b) leading an expedition in search of his father, (c) picking up a sidekick along the way, and (d) putting on a good show in interactions with Nestor and Menelaos, who are already acknowledged as leaderly. That’s in Odyssey books 2 to 4. Lter on it becomes clear that (e) he’s super-strong as well (he’s the only person other than Odysseus who’s capable of bending Odysseus’ bow) and (f) he’s capable of flyting, combative talk, just as much as Odysseus is (the closing scene of book 24).