To add to KennyG's point...many other wages are directly or indirectly indexed to the federal minimum. In other words, when the lowest paid workers get more, everyone above them wants (and gets) more because their work is more valuable than the minimum or entry wage. That rolls all of the way up and actually broadens wage disparity rather than the narrative that higher minimum wages shrink inequality.
SS, by contrast, is not driving up competition for wages. Workers cannot just stop working at any point and make more money on SS. There would be slight upward inflationary pressure, but it is minimal by comparison.