I agree that this question is better addressed on the history exchange. And while I also agree that the term "corporatist" is misused, the quote is not incorrect. It basically dispels the dangerous nonsense that equates Fascism and Communism, which is a bit like arguing that private land ownership is tantamount of feudalism and leads directly to Absolute Monarchy.
The idea that both forms of government assume totalitarian structures in contrast to liberal, market-based democracies is most powerfully argued by Hannah Arendt. While the contrast makes sense, the crucial distinctions were then elided by Austrian libertarians like Hayek and subsequently degraded into Chamber of Commerce agitprop.
In reality, all governments of advanced economies, including the United States have extensive state planning of the economy and welfare systems, which expanded enormously in response to the Great Depression. Moreover, all states at war (and Fascism seeks war as part of its innate structure) are effectively statist, in the sense of state domination of key economic sectors, at least at the "commanding heights" of the economy.
However, in socialist governments the corporate "means of production" are technically owned by the state and do not distribute stock shares or earnings to private shareholders. By contrast, the Nazis favored and pursued private ownership in a manner much closer to the United States than to the Soviets. Businesses and banks that had been publicly held under the Weimar Republic were privatized by the Nazis, and Hitler explicitly favored private markets as a way of "weeding out the weak."
Hitler could undoubtedly replace executives if he wished and exert power as the biggest customer in a war economy, while also providing slave labor. But he largely struck up bargains with corporations that retained private ownership, private management, and large private profits.
This is in stark contrast to socialists states. It is a partnership between private, capitalist industry and an authoritarian single party state. Tellingly, Hitler made it quite clear to the business elite that his own party's willingness to savagely suppress democratic elections and trade unions would protect businesses from taxation, public ownership, and other welfare measures advancing at the time.
In this sense, if I may opine a bit, the Nazi's elimination of opposition parties and its antidemocratic, antiunion partnership with private capital is closer to the GOP's revanchist ideology since the Justice Powell Memorandum and the "K Street Project" than it is to the Soviets or even Scandinavian socialism.
What characterizes Fascism, then, is not state ownership or domination of private industry, but a one-party domination of the state bureaucracy and the public sphere, which may be perfectly compatible with private capital. Arendt was correct to draw parallels between anti-liberal, antidemocratic states and the collapse of institutional divisions of power.
But the Nazis themselves were quick to round up and kill trade unionists, socialists, democrats, and even moderate left wingers. So describing Fascism as "leftist" is ahistorical, Orwellian nonsense on the part of the American right, and for the past several decades has provided talk-radio cover for what seems to be evolving at the fringes into explicit racial Fascism.