Though Doris Lessing was a member of the Communist Party during part of her life, there is, in my view, no special reason to interpret "The Old Chief Mshlanga" as embodying a specfically Marxist ideology. At the time the story was written (1951) much or most of the Western world was still lacking any understanding of the immense injustice of racial oppression in South Africa and elsewhere. The US South at that time, of course, was still governed by a legalized system of racial segregation. Many conservative people in the 1950s tended to associate liberal ideas about race with Communism, partly because the Communist Party was, or professed to be, in favor of full racial equality, while the mainstream political parties in most Western countries were not.
That said, it's clear that Lessing's description of the apartheid system in this story can be subjected to a Marxian analysis, which would tend to identify racial oppression as having an economic basis. The climax of "The Old Chief Mshlanga" occurs when the Chief's goats overrun the property of the young girl narrator's father. The goats are confiscated and the Chief's people are relocated—that is, run off their land, just as Native Americans were in the US. Beyond this, the point of Lessing's story is the complete cultural disconnect felt by most people of European descent with the indigenous South African and Zimbabwean population. The narrator herself recognizes how wrong this is, but in the context of the times, her sympathy with Mshlanga is an anomaly. The disregard by the other white people for the feelings and needs of the indigenous population is described in stark terms. Lessing's story, in describing this mindset and the conditions it produced, in South Africa and elsewhere, is a chilling reminder of just how deeply unfair, oppressive, and dysfunctional this system was.