The military runs as an authoritarian. You have your top guy, his team of generals, and down the line it goes.
Fascism: A system of government marked by centralization of authority under a dictator, a capitalist economy subject to stringent governmental controls, violent suppression of the opposition, and typically a policy of belligerent nationalism and racism.
Now we can pretend the USA is a fascist regime for sake of argument, but that does not mean that all armies of all history have been under fascist regimes. Stalin was most certainly not a fascist. Mao also was not a fascist. I don't remember the Japanese Emperors name but he was as the same suggest, the Emperor, not fascist.
Before modern economic systems, most armies were under the control of the Monarchy or Emperor or Republic.
Armies aren't controlled by economic systems. They are typically controlled by governments that belong to different types of economies, such as capitalism, socialism and communism. You can also be democratic, theocratic, authoritarian or some mixture thereof. How you choose your government doesn't automatically demand your economic system.