This is not a new argument and has been a right-wing talking point for as long as I can remember. I'm guessing it has become prominent lately because of the public visibility of white supremacy in the United States.
American conservatives often accuse Nazism of being a leftist ideology and therefore claim that liberals are closer to fascism than they are. Naturally liberals dispute this and argue the opposite. Of course neither is really correct. Nazism is neither conservative nor liberal - it is reactionary. That means that in its social policies, it is beyond conservative; however in its economic policies it is more moderate and has some traits in common with modern liberalism. However, liberalism is not socialism either, making this argument even stupider. Only in the United States would liberalism and socialism be conflated in that way.
The social agenda of fascism is obviously far-right and antithetical to liberalism. If we look at the economic ideology of fascism though, we see that it is center-to-center-right. If it appears leftist, that's because in this age of globalization and neoliberalism, socialism is no longer a major force.
Here's what fascism in general believes, economically:
Socialism is bad, communism is especially bad. The Nazis had a socialist element but purged them from the party in the Night of the Long Knives. They also hated the Soviets, possibly even more than they hated liberal democracy.
Free-market capitalism is bad, but not all capitalism is bad. Statist capitalism is good. Private property is alright, as long as businesses don't oppose the state. Globalism is bad (since it's a conspiracy by international Jewish bankers) and protectionism is good. Private enterprises and the state should collude with each other. (This is known as corporatism.)
Equality is bad and class conflict is a Marxist lie. Inequality is natural and good. Social classes should exist and everyone should accept their place in the hierarchy and work for the good of the state. This idea is anathema to socialists.
A welfare state is good, but only for whatever nationality/race is the dominant one. Note that a welfare state is only a leftist idea in America. In Europe, it's a mainstream, centrist idea and has been for decades.
Nazism and other kinds of fascism made conflicting, confusing statements about socialism and whether they supported its ideas or hated them. They also made similarly confusing statements about capitalism. They were consistent about hating communism though.
Based on their track record, I think it is fair to say that the Nazis and other members of the Axis were not socialists, nor were they free-market capitalists. They in fact didn't care that much about economic ideology at all. They were interested mainly in nationalism and warfare. The fascist economic position can be described as pragmatic. Their position is "we don't want a wholly government-run economy because that's Jewish Bolshevism, and we don't want a free-market economy because that's like liberal democracy, but we want a protectionist kind of state capitalism where big business and government are basically the same thing. We hate equality and love inequality. We think private property is alright. We're okay with a welfare state and public works, but only if it benefits the Aryan/Italian/Japanese nation. Most of our social spending should go towards the military though."
Most of these ideas (glorifying inequality and the military, maintaining rigid class structures, corporatism) are horrific to socialists. Likewise though, other fascist ideas like protectionism, welfare states, military-industrial complexes, public works and nationalization of select key industries are opposed by free-market capitalists.
Tl;dr fascist economics were a mess and borrowed ideas from both left and right as a syncretic kind of ideology. Fascists didn't really give a shit about left vs. right when it comes to economics and they hated both socialism and free-market capitalism. However, they thought a statist kind of capitalism was desirable where competition was allowed and classism was awesome as long as it benefited the state.
Conclusion: Fascism is not a leftist ideology. Socially, it is extremely authoritarian and far-right. Economically, it is more moderate and centrist or center-right. Anyone who tries to tell you it is socialist or free-market capitalist is a liar.