So after looking over what we know it became a bit clear to me the way the story mode will unfold and why.
The rivals are setup by polygon man.. he basically is a jealous evil jerk who knows to get them when they are at their weakest. He sets them up to battle.. and once one of the rivals is defeated he will jump in and try to take the remaining life out of the fallen rival. You will then have to save your rival.. and yourself.
Imagine evil cole saving fat princess or the other way around?
I hope for this.. wouldnt it be cool if by consuming your rival he also uses their worldly powers against you? Which brings up why there are many easter eggs and references. All that "content"
During the consumption of your rival they begin to turn into just polygons.. and polygon man becomes more and more detailed.
The Ending?
You become friends with your rival.. :-D