The American term "middle-class" means something far less specific than the original British meaning of "well to do, town dwelling, non-aristocratic business owner". In that definition, a "middle-class" person is never an employee (which would make someone "working-class").

The American use of the term reflects (whether or not it is borne out in reality) the achievement of a comfortable and secure lifestyle (relative to the current time period) by an employee. In a way, it's a more flexible definition than the British one. The post WWII economic boom was the first time that, en-masse, people who were not inheritors of significant educational or monetary capital or status could achieve that lifestyle.

However, the problem with the American definition is that it chooses to ignore how much that achievement actually has to do with class and other inertial societal constructs, perhaps because frank discussion of class effects is still considered taboo in American society, whereas older societies with openly acknowledged class structures can actually have that discussion.