Marx identified about five classes based on their relationship to the ownership of the means of production and their likelihood to support or oppose the people who owned the means of production.  All white people were not rich by definition - in fact,  Marx's working class in 19th century Europe was about one hundred per sent white.  One thing that modern day identarians (social justice warriors, Arlo Kempf's, etc) are not is Marxist.  If anything identity politics is an anti-Marxist perspective.