Does weapon rank affect stats in Max Payne 3 multiplayer

No, they do not affect the story. Some clues will have an effect on other clues, though, like meeting the same person for the second time.

To my knowledge, there are no particular benefits to dual-wielding the same weapon. The level 6 unlock serves pretty much the same purpose as any other unlock: To keep you from doing what you might otherwise do so that you have something to aim for. Cynical way of looking at "progression," but it works. Likewise there are no "benefits" to unlocked clothing items that would necessarily justify their not being available initially.

That said, I do tend to dual-wield the same weapon more often than not, because 1) As you said, you level a single weapon up faster instead of spreading it between two, which is nice just because you'll get the good stuff (lightweight, barrel upgrade, etc) faster. More importantly... 2) Two weapons that behave the same way can be easier to use than two that are slightly different. The obvious case is differing ammo capacity (try using 1911 and PT92 without extended magazine on the 1911), but even vanilla 1911 + Bull can be frustrating when one weapon's firing rate is different than the other. In order to maximize both, you'd have to have a very strange clicking pattern. Or in other cases, the damage dropoff may be worse on one gun versus the other.

So yeah, no strong benefit, just potentially easier to use, and another thing to unlock.