It’s not, but they are both still socialist. Fascism tends toward the model of strict government control while Marxism government ownership of the means of production.
Naziism (i.e., fascism as practiced by the Nazi Party in the thirties and forties) was hardly socialist. It exercised control over industry mostly by issuing huge government contracts. As you yourself point out, a central feature of Socialism was state ownership of industry.
Here’s a case study for you.
Look at the Quandt family. The patriarch, Gunther Quandt, founded the industrial conglomerate that included BMW, its best-known company. The family got rich during the First World War, richer during the Second World War, and still owns a controlling interest.
And they’re still around, and still rich, and still own the company, or at least a controlling interest in it.
And the companies into which IG Farben was broken up are still around, too.
Now, let’s look at an equivalent industrialist in Russia before the October Revolution. Or actually a large landowner, perhaps one of the nobility who owned huge amounts of land and farmed that land (until 1860 or thereabouts) with serf labor, and subsequently with impoverished labor who might as well have been serfs.
Where are those nobles and their families today? Dead, mostly, or reduced to nothing. Their land was collectivized.
Nazi-style fascism and Socialism (upper-case “S” Socialism, the real thing, not Bernie’s proposals to do things in this country that are pretty liberal, often things we’ve done before, like more progressive taxes and heavily subsidized higher education) are
fundamentally different. There’s some overlap, of course – they both had significant totalitarian elements – but they are diametrically opposed.
