Fascism is just socialism with a nationalistic bent.
Fascism was the name adopted by Stalin to differentiate his form of socialism from Hitler’s. The media was quite happy to oblige old Joe and copiously followed his direction always noting Hilter as a fascist and Stalin as, well, as a communist. To the highbrows, there might be a smidgen difference in the idea of “ownership” but the operative fact remains the rule was the same. Hitler was a racists, Stalin a racist and classist. Between the two there is very little difference - they were both narcissists with a megadose of meglomania thrown in. Which one did the most harm - answer, Stalin; Hitler was defeated physically and ideologically. Stalin continued to deform and distort humanity and had a mutligenerational impact.
Hitler made plenty of speeches of how he wanted Germany to be a classless society.
Perhaps so, but "fascism" is an Italian creation, coming from the Latin word fasce, or "bundle."
Various Italian revolutionary groups adopted the Fasci name prior to Mussolini, who organized the group Fasci italiani di combattimento.