Cyrus did not actually use the title "King of Kings", but rather a traditional Assyro-Babylonian styling - "I am Kurash, the King of All*, the Great King, the Mighty King, the King of Babylon, the King of Sumer and Akkad, the King of the Four Quarters".

*Literally, "King of [Kish](https://en.wikipedia.org/wiki/Kish_(Sumer))".

In Assyrian title, the styling "King of Kings" shows up sporadically over the Middle- and Neo-Assyrian era, but often in never-ending, tedious lists of styles like "Unconquered hero", "Most exalted priest", "beloved by the gods", and sometimes variants of it appears in names from the Akadian era onward, so it wasn't necessarily a main title. Its popularity among Persian rulers (styling themselves "Great King, the King of Kings, King of Many Lands..." and similar variants) might be attributable to its use among rulers of Urartu (Armenia), who had been at least partly subjugated by Media.

In any case, the styling comes from Shelley paraphrasing Diodorus Siculus:

Ten stades from the first tombs, he says, in which, according to tradition, are buried the concubines of Zeus, stands a monument of the king known as Osymandyas. At its entrance there is a pylon, constructed of variegated stone, two plethra in breadth and forty-five cubits high; passing through this one enters a rectangular peristyle, built of stone, four plethra long on each side; it is supported, in place of pillars, by monolithic figures sixteen cubits high, wrought in the ancient manner as to shape; and the entire ceiling, which is two fathoms wide, consists of a single stone, which is highly decorated with stars on a blue field. Beyond this peristyle there is yet another entrance and pylon, in every respect like the one mentioned before, save that it is more richly wrought with every manner of relief; beside the entrance are three statues, each of a single block of black stone from Syene, of which one, that is seated, is the largest of any in Egypt, the foot measuring over seven cubits, while the other two at the knees of this, the one on the right and the other on the left, daughter and mother respectively, are smaller than the one first mentioned. And it is not merely for its size that this work merits approbation, but it is also marvellous by reason of its artistic quality and excellent because of the nature of the stone, since in a block of so great a size there is not a single crack or blemish to be seen. The inscription upon it runs: "King of Kings am I, Osymandyas. If anyone would know how great I am and where I lie, let him surpass one of my works."

Therefore, the translation (which may or may not be an actual translation - it could be Diodorus making up an appropriate generic assertion or paraphrasing something he heard or read in Hecateus) derives from a Greek-speaker living in the Hellenistic era. So why would he use the styling "King of Kings"? That's simple - Diodorus was discussing Rammesses II, a legendarily powerful and ancient king. To Siculus, a 1st-century Greek, the Great King of Persia, along with Alexander and the Diadochoi, would have been the figures that defined what it meant to be a powerful, autocratic monarch. To say that Rammesses was anything less than a King of Kings would surely have been to understate the legacy of his rule.

You can perhaps compare this to how we in the modern day say that when the Wang (king) Zhao Zhen of Qin declared himself Qin Shi Huang Di, he became the first Chinese Emperor. What we're doing is simply comparing the European notion of an Emperor being superior to a king and ruling an institution that in some sense transcends mere kingship to Zhao Zhen's assumption of the title Huang Di, which you could arguably translate along the lines of "Divine King".