I think the problem with defining class in the US is that we did away with the formal concept of classes, so it's difficult to talk about.

We don't have a system of nobilities and honors in quite the same way the UK does.  So it's difficult to describe who the "upper class" is.  There is general agreement that if you are wealthy enough that you will never have to work, you are upper class, but it's often difficult to identify these people.  Someone might live in a desirable neighborhood, drive a nice car, travel, and wear nice clothes, but if they have to keep working to maintain that lifestyle, they're not likely to think of themselves as "upper class" because they didn't inherit their money and their lifestyle could all fall apart at any time.  Meanwhile someone else could live a very modest lifestyle without having to work and never think of themself as "upper class" because they're essentially a normal person who simply doesn't have to have a job.

There is general agreement that if you can't afford a place to live, enough food to eat, or a means of transportation, you're poor.  That's a pretty low bar, and it's rather difficult to fall to that level so long as you're trying at all, at least anywhere with a decent economy.  Instead we wind up with many people who do have a place to call home, who get enough to eat, and who can keep themselves mobile and working, but for whom practically any unexpected expense can threaten their lifestyle.  They may treat health and dental care as options of last resort, due to the added expense of accessing them.  They might never take vacation because time off is a luxury they can't afford.  These people are relatively poor, but do not necessarily think of themselves that way, because they are not so obviously different from anyone else who has to work for a living.

So most people wind up thinking of themselves as "middle class" because there's nothing very obvious to indicate when they're "poor" or "rich".