We're not using "money" because social class is partly perception-and as you pointed out, can be wrong. A middle class individual can "fool" someone into thinking their upper class by living deep in debt.
Keep in mind that most of the folks we're talking about when we say "lower class" are actually very poor folks. Not folks who have the credit to go into debt, but the folks who couldn't get a loan if their lives depended on it. Not folks who have a brand new BMW in the drive but no furniture, but people with barely a clunker and used furniture, if very much.
We're talking people who walk into the bank, and get immediately ignored. People who are already in debt *just keeping the lights on* in a cruddy apartment.
People who have the ability to obtain a mortgage are usually not lower class. They may be cash poor, but equity counts. It counts for a *lot*.
The people who have their new car but the crap food, the excellent address but no furniture, are most likely actually middle class with poor spending habits. Those are common enough, but not really representative of the majority. There are outliers-folks who don't manage money well, and folks who are trying to "keep up" for one reason or another.
If you took in their equity and then balanced it out with income and debts, they'd still be above those who are trying to raise a child on minimum wage with highschool education. If they weren't they couldn't make the car or mortgage payments (which combined are probably more than the minimum wage worker is earning or can hope to earn any time soon).
I also don't think this is the place to nitpick over terminology since the discussion is about whether being lower class and dirt poor can and does affect your mental illness, as opposed to upper class.
Yes-poor spending skills affect stuff, but that's not a measure of social class, that's a case of ignoring practicalities or a symptom of a mental illness of it's own.