It's very difficult to disentangle privilege/disprivilege and socioeconomic status, for sure, not least because there is causation in both directions (very high SES could lead a black person to overcome a lot of disadvantages that affect most black people; similarly, those disadvantages can lead to low SES).

Perhaps a useful rule of thumb would be to say that something is white privilege if a black person has to be EXCEPTIONALLY, not just averagely, successful in economic or class terms to be unaffected by it.