It sounds like you're talking about the barbarian fluff within drow society; but my point is that you can change the fluff, and easily at that, to fit perfectly within drow society.
You could say the same thing about the fighter--or about any primarily martial class that gets medium/heavy armor.
As for the civilized/barbaric aspect... again, it's fluff. You could easily fluff the rage as intense emotion, and not necessarily anger. It could be a deep-welled passion to protect allies that allows you to bypass the limits of your body for a short time which fatigues you. Or alternatively, a state of intense focus where you push your body to the very limits, causing your muscles to strain to the point where they hurt themselves (which again, fatigues you).
Just because the name of the class is "barbarian" doesn't mean they need to be barbaric, reckless, or anything similar. You mentioned Drizz't--he was defiant and reckless, and not in any way a barbarian. Likewise, a barbarian can be entirely obedient (within the limits of CE Drow society, of course, where almost everyone is CE), and only unleash that hallowed state when fighting the enemies of his House.
Finally, there's nothing stopping a barbarian from investing in stealthy skills to be just as quiet as any other fighter in light or medium armor. In that, he's potentially no different from someone who's taken levels in the Fighter class.
You can change the fluff however you like, and have whatever style of barbarian you want as long as you follow the mechanical rules. You can change the flavor it to suit the Drow without repercussions, and play what you want as long as you're open to refluffing the way the class works, which is far from set in stone, and nothing "bad" will happen if you do it.