Ah, Isaiah is talking about the devil, and when he rebelled. There are some similarities of 'roles' played in the flesh king of Babylon and the devil, but the flesh king of Babylon converted, whereas the 'spiritual' king of Babylon has already been condemned to death. The Heavenly Father had His prophets write down where the line in the sand is as far as how far HE will allow the devil to play his games on his way to being destroyed from within, gone forever.
In the allegorical sense you are, of course, absolutely correct, just as the same can be said about the Pharoah and multiple other figures. It’s important to remember, however, that the term used in the original Hebrew roughly translates to “Son of the morning star”, and that it is a title, not a name. It is also important to remember that the title is applied directly to Christ at times.
The other important thing to remember is that the devil is a usurper and always has been. That IS his sin. It is of little surprise that he would attempt to usurp a rightful title of Our Lord through his wiles and snares.