Yes, "like the wind" is a prepositional phrase functioning as an adverb, just like "up the stairs". It is describing how he flew, thus it modifies the verb "flew".
"Like" is definitely not being used as a conjunction. What would it be joining? There is only one person doing the flying -- "he" -- and he is doing only one thing -- flying.
As to "as" ... We generally use "like" in such a phrase when we want to say that the action performed is similar to some other action or the person doing it is similar to another person without actually being that other thing. We use "as" when we want to say that this is an example or characteristic.
"As a good citizen, you should vote." You ARE a good citizen, and good citizens vote. Therefore you should vote.
"Like a maniac, Bob ripped open the package." Bob is not actually a maniac, but in this particularly case he acted in a way that resembled the way maniac's behave or we expect maniac's to behave.
That said, "like" is often used in the sense of "as". This is sometimes blamed on one advertising campaign many years ago that used the slogan, "Winston tastes good like a cigarette should." The grammar Nazis of the time howled that it should be, "Winston tastes good AS a cigarette should." Winstons were not "like" cigarettes: they were cigarettes. But the usage caught on, so "as" is in retreat.