From a materialist lense, middle class usually refers to the small business owners, landlords, etc. Petty bourgeoisie basically. They historically tend to welcome fascist ideology out of fear of losing their privileged position in society.
So there’s a difference between the working person who might get caught in a false consciousness versus the tenuously well-off person who’s somewhat class conscious. The latter is likely a lost cause more often than not. The former can often be reasoned with if we can speak to their experiences as a worker and cut through the spectacle.
But yeah the Liberal use of the term “middle class” as someone occupying arbitrary income brackets is an immaterial abstraction with very little utility for either prediction or description.