When it comes to facism's invention, you need but look to the Eastern Roman Empire.
They developed the science and practice of fascism long before Modern Italy. In fact, they had figured it all out before Mohammad came along.
Islam grew up fighting against the Eastern Roman Empire for about 700 years. As they did so they adopted many of their ideas of how to govern.
Today's modern Islam is as fascistic as Byzantium ever was, and could have served as a role model for the Nazis.
It's true that the fascist party was started in Italy and came to power there under Mussolini c. 1922. But Hitler did borrow a lot of his ideology from Mussolini (e. g., Italian "Black Shirts" were the inspiration for Nazi "Brown Shirts," the economics of a "corporate state," extreme nationalism and military conquest). The two regimes formed the Axis alliance and fought side by side in many battles (especially in Africa) during WWII, despite the fact that Hitler and Mussolini did not always see eye to eye. In fact, Nazi troops rescued Mussolini from his first capture in 1944. So the "Allied propaganda machine" was not entirely off base in lumping the two regimes together and giving them the same name.