Well, communism is an economic system. Fascism is an economic system. Many nations have had the communist economic system or then a system with a high degree of communist elements to it, and all have been totalitarian. Italy had a fascist system once, and was totalitarian. Germany had a fairly similar system as well (although they never considered it fascist - that's just a perversion introduced by Germany's then enemies), and was totalitarian.
It doesn't follow that fascism would be communism. It just follows that totalitarianism is popular in all sorts of economic systems, ranging from fascism to communism to all sorts of free-market capitalism. And even that doesn't tell us much, because totalitarianism is popular on its own account. Only mercantilism would actually require pairing with totalitarianism in order to be an effective economic system. And conversely, totalitarianism is one of those forms of government that has good odds of working in company with an inefficient economic system (both in the sense that it can uniquely cope with the handicap, often through conquest and coercion, and in the sense that it can thrive on the economic suffering).
Whether Klingons have a class society or a caste society would depend on who is speaking. Nazis never had much of a say in whether their system was fascist or not, as the inaccurate expression (descriptive enough, but Germany would have insisted on a native name rather than an Italian one!) chosen by others has stuck; the US was and remains imperialistic to its enemies (and many allies).
So, we might choose whether to believe Antaak or Kolos on this. Kolos has antipathies towards the existence of the warrior class, so "calling it names" would make sense. Perhaps they really are a caste, near-divinely established, but Kolos feels they should be a mere class, breachable by mortal means? Antaak in turn speaks as the (past and wannabe) member of a caste, so "calling it names" doesn't make similar sense. Put together, the formally correct way to describe the Klingon society in the 22nd century thus might be as a caste system. But DS9 portrays definite class movement which clearly contradicts the idea of a caste system.
Timo Saloniemi