You could frame it as a class issue, but democrats have a much stronger hold over the African American voting population than the poor population. So it's better for democrats keep to framing conversations around race instead of class, to fire up their base.
I don't think the democratic party has a hold over the African American voting block in so much as it has been the only real choice for them for the last 40 or 50 years (unless you have Stockholm syndrome that is). Couple that with the increase in leadership roles and overall candidates and the democratic party is becoming more representative by the day. This results in more and better legislation targeting the issues of black Americans.
Contrast that with the republican party, how well are poor whites represented in the party (aside from near-constant lip service)?
The republican party platform combines the roughly orthogonal financial interests of the wealthy and moral interests of white evangelicals. Some of the key demands of poor whites are addressed within that latter category.
