I don't even understand why it exists. Same with Star Wars/Star Trek. Plenty of great stories and characters from all of these franchises, I don't want to choose a side.
I imagine it started something like this:
Ancient fan 1: I love all comics... but prefer DC a bit more.
Ancient fan 2: I love all comics... but prefer Marvel a bit more.
AF1: Oh that's nice. But tell me, why do you prefer Marvel more?
AF2: Well I like DC too, but Marvel is *reasons*. How bout you, why do you like DC more?
AF1: Oh I like Marvel too, but DC is *reasons*. Don't you think so?
AF2: Well, that's nice. And yes, DC is nice, but I have to disagree because *reasons*
AF1: I see your point, but respectfully, I have to disagree because *reasons*
AF2: Ok, ok... but regarding your point, I think you're wrong. Marvel is better because *reasons*
AF1: Now hold on a minute, DC is better because *reasons*
AF2: Woah, I think that's reaching a bit. Marvel is better because *reasons*
AF1: Hah! No way, man. DC is better because *reasons*
AF2: Sorry, not buying it. Marvel is better because *reasons*
AF1: Ok, now you're just being biased. DC is clearly better.
AF2: Who're you calling biased? You don't need to be a jerk. Marvel is *reasons*
AF1: Oh so now I"m a jerk just because I like DC?
*argument devolves into a catfight. Rinse and repeat over a few decades and that's how we got here*
Anyway, my point is that I'm pretty sure this rivalry started out as a very friendly chat about a few guys who love comicbooks stating which comics they liked better and why. Evolve that argument over a few decades, include quite a number of anti-social nerds into the mix who love their comics, and you can see how we ended up with this rivalry today.