Exactly. That said, some standardized tests have been accused of class or race bias merely by the demographic distribution of outcomes, which requires a lot more explaining than asking whether "Does Buffy's Jaguar leak oil?" has a potential class bias.
In the U.S. class, income, and race are correlated for various historical reasons. Obviously "correlated" means "on average", not a rigid relationship. But discriminating based on class is very likely based on correlation to produce a disparate impact based on race.
How disparate, and how bad that is and even I suppose whether it's bad, are hotly contested ideological questions.