Among the things you preserve by making sure you are not mistaken for a lower social class are:

* whether people invite you to social events

* how people treat you when you ask for help

* what kind of sex partners you have access to

* your own perception of yourself as worthy and accomplished

It's not an economic class thing, it's a social class thing and would exist even if everyone had the exact same income.

Meanwhile reaching for millionaire status when you don't really have it is stressful. Other people will push you back down and you have to fake a lot of behaviors you don't really know how to use. So people generally accept their level instead of identifying with a higher class. This is true in theory and also when you actually talk to them.