Within the Tanach/Old Testament there is no association of the angelic “adversary”, the satan¹ in the books of Job and Samuel, to be any sort of fallen or rebellious angel. Aside from the rather obscure verses about the nefilim in Genesis 6, I know of no Biblical verses that Jewish scholars take to refer to fallen or rebellious angels.²
The verses in Isaiah & Ezekiel, therefore, can be applied to some actual person—the king of Babylon in Isaiah, of Tyre in Ezekiel—who set himself up as a great power—“to the heavens” or “as a god”—and was/will be cast down.³
As such, I’m not sure this is answering the question “Why is the King of Tyre conflated with Lucifer?” so much as pointing out that this is a conflation, and not necessarily implicit in the text.⁴
I have written the word in italics to emphasize that I’m using it as a transliteration of the Hebrew שטן, as distinct from “Satan” as a proper name.
As swasheck points out in the comments above, the extra-canonical book of Enoch does describe these fallen angels in more detail.
As usual, it’s not always easy to determine whether a prophecy refers to a contemporaneous event, one being predicted for the near or distant future, or even one in the past (Ezekiel doeas this at least once, IIRC); it’s also ambiguous whether a literal king of Tyre/Babylon is intended or this symbolically refers to someone else.
Well, Isaiah does say “Lucifer”, at least if you translate into Latin; but a translation into English might go, “How art thou fallen from heaven, O day-star, son of the morning!” (JPS 1912).