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How come we have a bipartisan system in the US and how does the conflict between “Left” & “Right” arise naturally in so many cultures?

We need to understand that at the base those are two strategies that are driven by different behavioral strategies, it’s modeled in game theory by a game called Hawk vs. Dove. 
 
For our discussion let’s call it Wolf vs. Pigeon:
The pigeon behavioral pattern (Left Wing) is based on very well defined cultural norms and rules that enable cooperation between members in the flock. 
The Wolf pattern (Right Wing) is not taking for granted rules and social norms but assume the famous “Homo Homini Lupus Est” saing, meaning that to each his own under some profit loss economic and survival constraints. 
 
In both strategies there is benefit in creating groups, either flocks of pigeons or a pack of wolves. 
The pigeons create a flock, motivated by self protection, better foraging and energetic efficiency. 
While the wolves create a hierarchical pack to hunt better prey and to reduce the risk of self injury from other canine teeth carrying members of the pack.
 
We see that the main difference between the pack and the flock or Right Wing and Left Wing is the concept of Hierarchy. The Left Wing is bound by implicit social rules so the flocking is based on equality assumption while for the Right Wing the packing “order” is hierarchical because, otherwise there is chaos. 
Democratic order is a mechanism based on equality of voting used to create a hierarchy of political representatives, so the difference between how Right and Left Wings see this mechanism is the emphasis of the Right on the hierarchical structure and the Left on the equality of the representation process. 
 
Left Wing members will always be pulled to processes that stress ongoing control of the social power and hierarchy, while Right Wing members will stress order and conservation of the hierarchical system. 
 
At the extremes we can see the Anarchists vs. the Fascists point of views modeling this theory, but it’s just an individual selection of the member into which flock or pack he feels that she belongs.
 
In democratic processes humans make the process more complicated because some wolves have to be elected by doves and vice versa, so one gets more strategies in the game like: wolf-like pigeon and pigeon-like wolf, and at the end total dislike from political representatives, and the common saying that Democracy is necessary evil. 
 
Actually this is exactly the case – Arrow’s paradox that was defined by the economist Kenneth Arrow in his 1951 book “Social Choice and Individual Values.” states mathematically that there are inherent difficulties in creating a perfect method for aggregating individual preferences into a socially optimal outcome while satisfying a set of desirable criteria – i.e democratic decision process is flawed by nature.
 
That is of course till AGI (Artificial General Intelligence) will decide for us all.