So the source of confusion here is that these aren't companies, like Boeing or Lockheed were. They were State Design Bureaus. But your observation isn't totally off the mark. Pre-war, Tupolev existed for Bomber design, and Polikarpov existed for fighter design, and although others existed, those were the primaries ('Internee Design Bureaus' existed for designers who were under arrest to continue work under much closer supervision). The Tupolev/Polikarpov monopoly didn't last forever though. Eventually more Bureaus were created or expanded to leverage the talents of various designers better, which meant that Mikoyan, Yakovlev, and Lavochkin now were fully in the military aircraft business, starting in '39. Just in time, too, since it meant the designs were just getting set for their debut when war broke out with Germany, and the earlier generation of aircraft proved to often be well outclassed.

So anyways, the point is, the design Bureaus basically represented talented head designers, and while early on there wasn't much overlap, this changed to better maximize their abilities, but it shouldn't be seen as "competition" in the free market sense.

The Soviet Air Force Since 1918 by Alexander Boyd is a bit outdated, but covers the developments in that period well enough.