Benito Mussolini was the first leader to lead under the “fascism” banner. Having placed himself there, he offered the core definition of fascism: “All within the state, nothing outside the state, nothing against the state.”
You’ll notice that Mussolini did not claim that the state owns everything within its borders. He simply notes that the state is the overarching institution, controlling everything.
Following this fascist credo, Hitler did not nationalize German business. Instead, the Nazis allowed private ownership, with private profits, provided that the government called the shots.
Indeed, the simplest definition of fascism is contained in that last sentence I wrote. Fascism is a system that ostensibly allows individual liberty and private property, provided that it’s always the government that calls the shots.