The "like me" here points in multiple possible ways.
I've taken to interpreting it as an invitation to examine the degree to which I personally do the thing being critiqued.
If I don't do it? Then yeah, it's middle class white men "like me" structurally, but not "like me" individually.
If I discover, to my horror, that I actually do the thing? Wow! What a gift to be called in and be given the choice to do better!
White people like me, we all need to do better.