Read carefully, the government still had ultimate control over the businesses. Hitler, Mussolini and others repeatedly tell you they are socialist. It's still socialism, just a nationalistic socialism.

“‘Private property’ as conceived under the liberalistic economic order … represented the right of the individual to manage and to speculate with inherited or acquired property as he pleased, without regard for the general interests … German socialism had to overcome this ‘private,’ that is, unrestrained and irresponsible view of property. All property is common property. The owner is bound by the people and the Reich to the responsible management of his goods. His legal position is only justified when he satisfies this responsibility to the community.” —Ernst Rudolf Huber, official Nazi Party spokesman, 1939

“I have learned a great deal from Marxism, as I do not hesitate to admit. I don’t mean their tiresome social doctrine or the materialist conception of history, or their absurd ‘marginal utility’ theories and so on. But I have learnt from their methods. The difference between them and myself is that I have really put into practice what these peddlers and pen-pushers have timidly begun. The whole of National Socialism is based on it. Look at the workers’ sports clubs, the industrial cells, the mass demonstrations, the propaganda leaflets written specially for the comprehension of masses; all these new methods of political struggle are essentially Marxist in origin. All that I had to do was take over these methods and adapt them to our purpose. I had only to develop logically what Social Democracy repeatedly failed in because of its attempt to realize its evolution within the framework of democracy. National Socialism is what Marxism might have been if it could have broken its absurd and artificial ties with a democratic order.” — Adolf Hitler