That's a really interesting point that I've always wondered about, and I hope someone here knows more than I do about using socioeconomic background rather than race as a factor in admissions.
From what I understand, affirmative action is more about a person's race and gender. Therefore, an upper-middle class black, female applicant with application X would be much more likely to gain admission into College Y than a poor male white applicant with the same qualifications. I don't think anyone would argue that the former applicant has more resources to assist her, and likely had a more stable home life. Furthermore, the quality of education this girl received was likely better than that of the latter applicant, because her family can perhaps afford to send her to a private school, or to a prestigious boarding school.
In my eyes, if the relatively disadvantaged applicant was able to keep up with the well-off applicant, it would suggest to me the latter applicant had to put much more work in to achieve the application parity, or is naturally more gifted than the other applicant.
By this logic, it seems that, in a country where no one race has an absolute monopoly on poverty, and background is much more imperative to achieving equality and fairness in college admissions than race.
But I honestly am not fully informed, and I would appreciate it if someone could enlighten me.