First of all, historically the middle classes referred to the bourgeoisie who were in the middle compared to the landed aristocracy. Nowadays it means the petite bourgeoisie, e.g. small business owners, learned professionals, middle managers, who are in the middle compared to the bourgeoisie.
The idea that "the middle class doesn't exist" is, as far as I know, a misunderstanding of Marxist class analysis. Marxist class analysis isn't really about defining a label for each individual, and classes are often reabstracted, re-split from society, differently depending on the specific goal of the analysis. It is correct to say that "modern society consists of the bourgeoisie and the proletariat (no middle class)" when it is appropriate for understanding some phenomenon, but incorrect to claim someone's analysis is wrong because they used the concept of the middle class, or any class for that matter.