Really anything 2 people would fight over would work. What made the cold war cold was that these two forces couldnt fight eachother without risking MAD (Mutually Assured Destruction). As long you habe conflict, and some means that both could do that woukd wipe out the other, then you have a cold war (till one says screw it and "pushes the button").

You could have as you mentioned capitalism vs communism, you could do warring family's, different forms of government, a conflict that started over a stolen ladder and who owns it, a piece of land, goblinoid vs humanoid, magic vs nonmagic, really anything (no matter how small) can work just let it spiral a bit and boom you have a conflict.