Communists use the term “Fascist” to define anyone they disagree with.
Notice they always referred to Nazis as “Fascists”, even though the Fascists were from Italy. Why did they not call them “Nazis”? Because Nazi stood for “National Socialist”, and that would be admitting that they really were fellow Socialists.
The Red’s real complaints about Nazism were it was nationalist thereby undercutting their workers of the world nonsense, it did not destroy the corporate structure it controlled it while allowing profits, it allowed private property, and it provided the “benefits” of socialism, medical care, housing education without the collectivization of the Reds thereby making it more attractive to many more people in society. Roosevelt’s New Deal drew far more from Italian fascism than it ever did from Red communism.
Yes, your point is correct. The Fascists were, at most, center/right totalitarians. The actual right in most of Europe were the Monarchists. And the Nazis, unlike the Fascists, openly blocked the monarchial revival in both Bavaria & Austria. And note that the last Kaiser died in 1942, still in exile in the Netherlands.