Isn't collaboration between public and private sectors the essence of Fascism?

No, collaboration between public and private sectors is not the essence of Fascism. Fascism is an authoritarian political ideology characterized by an emphasis on nation and often race, as well as At its core, fascism seeks to organize a nation according to corporatist perspectives, values, and systems, including the political system and the economy. This includes an indication of the state's authority in the economy, labor relations, and the social fabric, among other areas. One of the aspects of fascism is the idea of a paramilitary youth organization, which would ultimately be responsible for instilling a sense of discipline and loyalty, and for reinforcing the state's authority.