This does not make them middle class. They are choosing to live in the expensive area and thus paying heavily for it. They could move 20 miles outside of the city, commute longer, keep the same high paying job and live in a mansion. The fact that they have a choice is what makes them not in the middle class. They see value in paying the extra cost to live in the expensive area and thus they choose to pay the premium.

People in the true middle class would not have the luxury of choice. They have no option to live in the wealthy area because they are not wealthy.