Yes, upper working class folks tend to describe themselves as middle class when surveyed. Partly through aspiration, partly through embarrassment. The extent to which they describe themselves as middle class is not the same as the extent to which they believe themselves to be middle class.
Meanwhile, at the other end, the petty aristocracy of the USA like to define middle class as anything below a million a year. I've seen recent posts describing people who own multiple houses as "lower middle class".
There are ways around the issue. Try "ordinary Americans". Try "working Americans". Try "the American people".
But they need to stop pretending that the working class don't exist and that this ambiguously defined "middle class" are the sole deserving beneficiaries of all political largesse.
Median household income in the USA is about $55,000. Households vary, but that's probably an average of about $35,000 per adult once you account for single parent households and 3+ adult houses.
Half of America makes less than that.
If you're pitching your policies at people who can reliably afford to buy houses, you're missing most of the people.
The reason why centre-right "liberal" parties like to pitch for the middle class isn't because everyone identifies as middle class. It's because they know that the working class have nowhere else to go when there isn't a genuine left alternative. A far right vote is suicidal for workers, and most of them know that.