The literal meaning of privilege is "private law". An example is OJ Simpson buying a not guilty verdict in a case where poor people of all hues would have been found guilty and rich people of all hues would not.
Intentionally or not, privilege theory erases class. For example, when you include class, you find that poor white and black people are equally likely to be killed by the police, and middle-class white and black people are equally unlikely, which means poor white people do not have white privilege. And suggests that if race is your only tool of analysis, middle-class black people have "white privilege" in this case.