The Nazi Party, even though it has "socialism" and "workers" in its title, is not considered to be a form of Socialism (nor, for that matter, to be pro workers). Not that it matters much, but the only mention of Nazism in the Wikipedia article you linked doesn't refer to it as a form of socialism.
I would not call them socialist exactly but the philospical background for both ideas is quite simular.
This is history that is mostly ignored or simply pased out of memory. Their was a version of socialism that was called 'Prussian Socialism' and the thinkers from this area had quite a bit of influence on the later devlopment of facism.
Many facists like mussolini were socialist that were pissed of by the socialist party. Hitler took a party with socialist elements and retracked it so that the socalist parts were neglected.
If you look at the original 23 points of the Nazi party, they look fairly simlular to many moderet socialist parties of the time.
The first chapter of Hayek 'The Road to Freedom' is an exploration of the history of thought. It was originally be intended to be part of a book on the subject (The Use and Abuse of Reason Part 1) but he had to no time and so he released it as a more populist book. Sadly this chapter did not get the attention needed and was removed in the version that became a best seller in the US (Readers Digest Version).
The Nazi Party, even though it has "socialism" and "workers" in its title, is not considered to be a form of Socialism (nor, for that matter, to be pro workers)
I hate getting down in the weeds on this, but its "not considered socialism" by those on the left who are embarrassed that the Nazis were pretty much a command economy and have socialism in its name.
I don't think Nazism is considered a form of Socialism by pretty much any reputable source; only some in the anti-Left fringe do. A planned economy alone is not enough to classify a system as socialist, and anyway Nazism had enough elements of private property and corporatism. The Nazis, like in many other aspects, were equivocating by calling themselves socialists -- not that they particularly liked the term themselves very much.
I wouldn't pay much attention to the name of the Nazi Party, either. After all, North Korea is a "democratic" republic, it says so right in their name!
So if a command economy isn't enough, perhaps it has to actually represent a form of community control, do you believe the USSR was any more socialist? What makes something actually socialist?
The understanding I've developed of socialism is of worker ownership of the means of production. The state at best can only claim to somehow speak for the workers in a top down manner, and I think that inevitably fails. (Which is not to say I'm advocating it here in a decentralized form but I'm much more sympathetic)
Frederick Hayek makes this argument in his famous book, The Road To Serfdom. Fascism isn't the capitalist response to socialism, it's the right-wing version of socialism. In Germany, it only came about because socialism had wiped out [classical] liberalism.
I don't think Trump is a murderous totalitarian (though I guess he could be), but fascism is actually an accurate label for his political ideas. He doesn't have a problem with state authority, he just thinks he could do it better, and would encourage nationalist social upheaval instead of SJW social upheaval. This is why the conservative movement hates him.
That communist countries often call themselves democratic is not really that strange. Democracy comes from the Greek words demos, which means people, and kratos, while means power or rule:
