I was too quick to write off INTJ for the narrator. The dry humor is very INTJ. I don't know INTJs well enough to know if the angst is, but maybe so. If it is, it's rarely on display like this. I haven't read the book, but in the movie, the narrator apparently has no clue what he wants - he's still dumbfounded by the extent of Tyler's creation, even in the end. INTJ still doesn't sound right, at least in the movie, because of the cluelessness and angst.
It would make sense that the narrator would create an ESTP as a charismatic alter-ego, but that defeats our argument, because he's not going to get it right, or blow it way out of proportion. For sake of argument, though: Tyler's dress is an anti-style, consistent with his contrary take on everything. From the 70s disco style on the plane to the dirty, ratty 1950s women's bathrobe he probably picked up at the Salvation Army. He wears other people's once fashionable, but hopelessly out of style discards. I don't see it as a way to say "look at me" as a statement of "I can say and do whatever I want, while you are conforming to societal norms" You could even say it is functional to his purpose. It's also an ENTP's nature to be contrary.