Olympic Gold, Silver and Bronze medalists in a 400m race all run fast.
Gold medal winner runs fast.
Silver medal winner runs fast.
Therefore
Gold medal winner runs as fast as Silver medal winner. (?) That would be a tie, and both would be awarded a gold medal.
(1) He runs as fast as I run fast..
(2) He runs as fast as I do..
(3) He runs as fast as I..
The underlined expressions are called comparative clauses. The crucial property of comparative clauses is that they are structurally 'reduced' in some way, usually obligatorily.
The second "as" is a preposition which is in construction with the first "as" (called the 'governor') which is an adverb.
In (1), there is no reduction and consequently the sentence is unacceptable. In (2) the comparative clause has been reduced to just "I do", where "do" is a pro-form meaning "run fast", so it now becomes acceptable.
In (3) the comparative clause has been reduced still further to just the subject of the clause, a pronoun. But "I" is very formal and best replaced by "me":
(4) He runs as fast as me".
Here, "me" is not a reduced clause, but a pronoun functioning as object of the preposition "as".
But, in some grammar books, the second "as" is thought to be a conjunction. Is it? Thanks!
Yes, traditional grammar analyses the second "as" as a conjunction because it introduces subordinate clauses.
But modern grammar rejects that analysis and calls it a preposition. After all, it is no different syntactically to the "as" in my example (4).
In other words, it's just a matter of complementation: it makes no sense to call "as" a conjunction when it introduces clauses like "as I do", but a preposition when it introduces noun phrases like "him".
I wouldn't worry too much about the different word categories. The most important thing is to understand the structure of comparative clauses, in particular the obligatory 'reduction' that I mentioned.