This is going to be pretty confusing when we go back to thinking that red symbolizes commies.
The swoosh communicates more densely, I think: it's a single, monochrome visual element, while the Obama logo is busy being a shape and a letter and a picture and a flag. The Nike logo says a lot, laconically; the Obama logo says a lot by being visually chatty.
There isn't anything intrinsic about communism that says "red, that's the color!". It was just a useful convention.
I don't know how much the Nike swoosh can say, perhaps movement/agility?. It's always been fairly meaningless to me, aside from the whole being the Nike logo thing.
"We" are a very small minority. Everyone -- the US included -- knows who Reds are. Only Americans, and folks who follow American news fairly closely, know what "red states" are.
China flag: http://www.worldclassflags.com/productimages/cf_china_rep.gi...
USSR flag: http://www.worldclassflags.com/productimages/cf_ussr.gif
Vietnam flag: http://www.worldclassflags.com/productimages/cf_vietnam.gif