Schopenhauer has had a profound influence in several different (but convergent) ways.
First, he was one of the first, if not the first, proponent of the idea that much mental activity is unconscious and that our conscious lives consist in large part of our rationalizing and living out unconscious, largely bestial forces. For this reason, he helped bring psychoanalytic thought into existence.
Second, he was a ruthless critic of what he referred to as ‘Christian ethics’, by which he meant an ethics of altruism and ‘turn-the-other-cheek-ism’, his position being that it was unnatural and would lead to the doom of those who actually embraced it. In relation to this, he was one of the first to suggest that Christianity is a bad psychological fit for people of European heritage. (Not that he believed it to be a good fit for anyone else, but his focus, for obvious reasons, was his own society.) Interestingly, this position was arrived at, sometimes independently and sometimes on the basis of a reading of Schopenhauer, by a number of extremely high level Jewish intellectuals, including Sigmund Freud and Maurice Samuel, and also by White Nationalists, including Nietzsche and Adolph Hitler. All of these people explicitly said that Christianity was fundamentally anti-European in nature. They also said that the people who did the most to promote Christian values those who believed themselves to have rejected Christianity—secular humanists, agnostics, and the like—-these being people who had rejected only the most superficial aspects of Christianity and had accepted wholesale the underlying morality of self-effacement. (This position is not exactly a stretch, given that the central tenet of Christianity seems to be ‘turn the other cheek’, since ‘the meek shall inherit the Earth’—-not exactly a recipe for virility and self-respect.)
Third—and this relates to the last point—-he was the first major critic of political correctness (which, as his writings indicate, was already alive and well in his time) his position being that it emasculated men and psychopathized women. This is another conceit of his that would be adopted and developed both by prominent Jewish intellectuals, usually in the psychoanalytic tradition, and also by White Nationalists.
Both Sigmund Freud and Adolph Hitler said that Schopenhauer was their ‘favorite philosopher.’
Fourth, Schopenhauer was one of the first—-and, to this day, remains one of the greatest—misanthropic, individualistic loners. Obviously there have always been misanthropic male loners, but Schopenhauer articulated and validated the values driving such people. As a result, such people are less likely to waste their lives as vituperative malcontents and more likely to look for constructive social and political outlets for their grievances.
Fifth, Schopenhauer was a hardcore critic of Universities and of state-run education generally; and because of the obviously Promethean proportions of his own intellect, his complaints carry a lot of weight.
Sixth and finally, Schopenhauer was one of the first, if not the first, to state and develop the idea that societies, being expressions in large part of human biology, tend to be structurally similar to one another, with the differences between them often proving to be similarities in disguise.
Schopenhauer is very much an intellectual alpha male’s philosophers and appeals enormously to rebellious male loners. I don’t know that he appeals, or had any intention of appealing, to other demographics.