Despite this being a cover (well, lyrically), it's pretty typical of what I've heard of him: he sings pretty folk songs much like the many many individuals regularly highlighted by Performing Songwriter magazine, but with goofy lyrics much like They Might Be Giants. Also like some TMBG songs, there's angst packed into a lot of the tunes, so that, for instance, a love song aimed at a laptop or from the point of view of a super-villain is still a love song, and in fact the lyrics can make this kind of song more palatable to emotionally skewed people like myself who might find a straight version too sappy or just plain ordinary to stomach. So he sings entirely with a straight face, with generally more subtlety than, say, Weird Al, but he still definitely produces novelty songs with more-or-less straightforward jokes, unlike the weirdness of Pavement or Robyn Hitchcock that in most cases still counts as rock (or in the case of Hitchcock, surrealist ranting) rather than comedy.
To bring in Goodman here, of course "Baby Got Back's" success as art and humor (if it's not obvious, I think these are related at the very least) depends on cultural associations, specifically the disconnect between the styles of the music and lyrics involved. Since it's a nice-sounding tune, one could enjoy it without knowing the source material specifically, though someone who didn't know anything about rap lyrics at all would likely just be mystified. Likewise, you don't need to know Coulton's style or his other work to get the song at all in (I think) nearly its entirety, though I may be overstating this: does knowing that he's an established, currently trendy musician with many such dishes on order make you enjoy this more than if you thought it was a one-off tune by some hack (like my "Billie Jean")?
Coulton, for me (having only been recently been introduced to him; maybe with additional listening I'll have a thorough and honest affection for him), is balanced on the edge between those bands that use humor as part of expression, which I think is good, and novelty bands, which I find generally dreadful (though, admittedly, I did go through a "Weird Al" phase a couple years ago; he's got some transcendent moments, though again, that's in part because the source material is actually pretty great, and his removal of the original lyrics takes out a barrier to my enjoyment).
It's a matter of whether it's sheerly parody, which is of typically very temporary enjoyment, being stale as the joke becomes old, or an actual, usable expression (to try to use Goodman's approach, here) that employs humor just because we, his audience, employ humor as an important element in our approach to life. If a joke doesn't get old, then it wasn't merely a joke in the first place, but some art activity that references the act of joking and/or some particular mode of mockery. This is again why I feel like "Mystery Science Theater" transcends humor-as-individual-jokes to become humor-as-life-approach, where free association and creativity rule (which has its dark side too, of course, for it can't be shut off even when inappropriate).
If you ever say (or think) "That's what she said" any more, it can't possibly be because you think it's actually funny, but it's become a trope for twisting whatever it is you don't want to respond to directly in the sense the speaker intends, and this in fact even works better, in my opinion, if it's a complete non-sequitur.
OK, since this is line is either going to devolve into an essay on "the tragic psychology of Michael Scott" or some sort of revelation of my individual pathologies in this respect, I need to stop now.