Despite the obvious logical problems (the excluded middle principal, the false dilemma), most "two kinds of people" observations are patently false for practical reasons. You're either a Beatles person or a Stones person (or Beatles/Elvis)? Really? Dog person or cat person? Even if you forced people to pick a favorite in these situations, you're ignoring those who hate pets. Or Amazon tribesmen who don't know what the fuck a Beatle is. You're only going to get it right if you claim that there are two kinds of people in the world: those who are identical to me, and everyone else. I suppose you could also claim that there are approximately 6.70 billion kinds of people (as of March 2008), and they correspond to Earth's population in a 1:1 ratio, but you'd have to meet them all to define your types.
The real purpose of the phrase is to establish an adversarial, "us vs. them" relationship. Flaccid one-liners aside, the speaker is usually on one side of an imaginary divide. You can love both the Beatles and the Stones, but some people are dedicated to one being superior, and that is somehow tied up in the existence of a diametric. The either/or thing is exclusionary... it's fightin' words. That's why Chuck Klosterman's claim in his essay "33" that "there is no relationship that isn't a Lakers-Celtics relationship" gets so much play in the sporting press. Because the Lakers-Celtics dynamic is so clearly based on an exclusionary relationship, people will convince themselves of its universality. It's funny because it's true! But Klosterman's point (and most other successful "two kinds of people" arguments share this quality) isn't that you either love the Lakers or you love the Celtics. That wouldn't be a terribly compelling read. His point is that a certain worldview is shared by the Lakers (or cat or Beatles or beer) person, one that is simultaneously opposed to and dependent upon its "opposite" number. And the people who buy into this argument do so because they have already bought into it. Everyone else just stops reading because the whole "debate" strikes them as pointless, which, ultimately, it is.
Which is why, platitudes, sad witticisms, and logical fallacies aside, it is true that there are two kinds of people in the world, Captain Kirk people and Captain Picard people.

Interestingly, the notion of Kirk and Picard as an either/or proposition is even more widespread (on the Interwebs, anyway) than the tired old "there are two kinds of people" idea. A Google search of the phrase "Kirk vs. Picard" yielded an impressive 64,000 results. By my (totally untestable) estimation, that comes out to millions of words worth of nerd vitriol. There was an official Kirk vs. Picard contest that allowed fans to submit projects supporting their fictional captain of choice. There are countless lists detailing Kirk's superiority, and just as many lists going the other way. This is decidedly not a rejoinder in that ongoing debate. (It's a stupid debate for several reasons, not least of which is the fact that both men had the same job, the same mission, and the same employer... and they even worked together once and seem to hit it off quite well.) The beauty of the "two kinds of people" argument is that it does not require that either side desire the subjugation of the other. The Celtics and Lakers are supposed to try and beat the shit out of each other. Lakers people and Celtics people, on the other hand, should be able to coexist peacefully (as long as there isn't a game on TV).
The same is true of Kirk people and Picard people. The two men are archetypes, and choosing between them is both unnecessary and fruitless; depending on where you fall on the nature/nurture debate, it's possible that we're all one or the other from birth anyway. You know the stereotypes: The Kirk person is passionate, loyal, lusty, brash, and quick to action. The Picard person is refined, noble, dedicated to reason and negotiation. The Kirk person likes sex, beer, and shirtless fighting. The Picard person likes romantic poetry, wine, and fencing. Kirk is fiery, Picard is sensitive. Put in the most flattering terms, Ernest Hemingway is a Kirk, while Noam Chomsky is a Picard. In less-than-flattering terms, Matthew McConaughey is a Kirk and James Lipton is a Picard.

Hell, if you were to just read down that list of characteristics, chances are I described half the marriages you can think of. (Chances are Kirk is the guy in, like, 90% of them, but that's another essay.) The supposed "opposition" of Picard and Kirk really breaks down to differences in approach and temperament, and only those who identify with one or the other in the first place feel the need to choose.
It will always be true that there are two kinds of people in the world, Captain Kirk people and Captain Picard people. Luckily, there's room enough in the universe for all of us.