That is a developed social hierarchy, not a natural one. An example of a natural social hierarchy is the belief that whites are naturally a superior race of that women are inherently subordinate to men.
Let me simplify. You used the example of Democrats believing in haves and have nots. Observing that some people are more wealthy than others is reality, not belief in a hierarchy. Believing that rich people are rich because they are innately better and poor people are poor because they are innately worse is a belief in a natural social hierarchy.
Close but not quite. Every human society has social stratification, even if it's only things like the jocks and nerds in high school. That makes it fascistic is the belief that those strata are innate and not social constructs. That is, the poor are inferior while the rich are superior, or whites are superior while blacks are inferior.