That'd elude most Americans, just like most of the world used "Russia" instead of the Soviet Union. The only real difference between those examples and, say, Persia or the Third Reich is that opposition to them is political rather than ignorance. (Albeit as stated, the Third Reich was an official name and not the everyday name used)
You forgot that I've been one of the chief explainers of the SSR system on this forum. The Russian SFSR was not the Soviet Union. It played a vital part (obviously) and was in many ways the center of power, but it wasn't the Soviet Union. I know that Russia already exists as its own possible revolt nation, but I'm using it as an example.
Actually if anything it's more ideologically charged. Bolshevist Russia to my knowledge basically went out of style by the 30's and when one said Russia, you basically knew they weren't aware of what the Soviet Union was. The debate between Iran and Persia had to do with the direction of state culture, relations with Germany, and relations with Britain.
I'm from Russia, I know what RSFSR was, and why Russia has an independence day
The point is, however, there is no in-game reason to choose Persia over Iran, so the choice must be arbitrary. But there is a real reason to choose Soviet Union over Russia even *if* you accept the reasoning that Persia should be used as it was more popular. You can't have two nations with the same name.
However, I should correct you on the Reich thing. The official name of Germany from 1871 to 1945 was simply Deutsches Reich - the German State. Certainly no one ever officially called it the Third Reich. Wikipedia, btw, claims that the name was officially changed to Grossdeutsches Reich later on.