Class in the US is absolutely an aspirational thing, but he's referring not to "class" in the literal sense of positional status within society, but more to the

Class in the US is absolutely an aspirational thing, but he's referring not to "class" in the literal sense of positional status within society, but more to the material trappings of "middle class life": things like a house, one or two cars, the ability to take occasional vacations, cable TV, an iPhone, etc. Arguably many or most people want many or most of those things because of perceived status advantages they provide, but that's another issue altogether.