It’s true that some corporations did very well under those regimes, but it’s not accurate to say that enriching corporations is why they did what they did. Hitler never thought “starting a war is the perfect cover for making my buddies in the private sector rich”, he wasn’t angling for a lucrative post-political career as a board member or a lobbyist or after-dinner speaker.
Corporate interests were pretty low down the priority list in those regimes. German corporations would certainly have been more competitive and profitable if they had kept their Jewish workers and been permitted to use "Jewish science". The Nazis merrily hamstrung the corporations in their territory for the sake of their overarching ideology, they totally subordinated economic interests to serve the State.
As far as the implication you mention, I wouldn't say that calling yourself a socialist automatically makes you one; I would say that to determine what "socialist" is supposed to mean it is informative to look at what people calling themselves "socialist" believe; but I would especially say that it's unlikely for the "national socialist workers' party" to have come to power by opposing "socialists".
I am not too familiar with all of this, but surely it's not such a weird thought that a party might call itself socialist even if it opposes the actual socialist? A bit like how a despot might call his party or country democratic.
The opposite of socialist isn't capitalist; it's individualist. Nazis were not interested in promoting diversity in society. Exactly the opposite: conscious and explicit elimination of diversity.
The essence of fascism is teamwork. A feeling of togetherness is its binding emotion, subsuming your identity into something bigger than you. (Only by understanding this can you understand why it's so appealing.) The bundle (fasces) that's tied together won't be broken. The problem is that whomever wields the bundle now has outsize power, and whatever is not part of the bundle is garbage to be swept away.