There's different "levels of privilege". Hanners clearly has access to some resources beyond what she makes at the coffee shop. Given what we've seen, she apparently makes little use of them. I can see Dad giving Hanners some stock or a small puddle of cash as she boards Shuttle to go down to Earth and when she gets there, she puts it into an account she rarely accesses; witness the checkbook she uses occasionally.
I follow a similar scenario in my own life. My wife and I worked as contractors for a decade. Early on, we put the clamps down on our spending and saved up a years living expenses. That was the level of personal safety net we felt we needed in case contracting went bad. First off, being able to save that much indicates we had relatively good paying jobs; and we still do 20+ years and several different employers later. Second, that we actually saved that much (and have kept it up to date as our expenses changed) says something about our ideas of living simply and within our means, and our thinking about savings. So thirdly, as a result, if something bad happens, the likelihood is we can afford to pay our way out of the situation. That makes us "privileged" in many peoples eyes, even though we are nowhere near "rich".
Example: our furnace failed one year while we were out of town. Got back and it was colder in the house than it was outside. Frozen pipes everywhere. I ended up replacing two toilets, two faucets, a dishwasher part, and redoing a bunch of plumbing and drywall in addition to paying for the furnace repair. Total cost: ~$1000 and 6 weeks of after work time spent on fixing things; it would have been nearer $5000 if I had to contract that work out, but we're both handy. (Fortunately, when I redid the plumbing in this house shortly after purchasing it, I put valves everywhere so I was able to isolate individual rooms with plumbing issues until I could get to them.) Some friends of ours had something similar happen the next year and they had to deal with their insurance company because they couldn't afford the repairs they needed otherwise. So are my wife and I "privileged"? You bet we are, even though we'll never have a second or third house or a sports car or ... And we got there by making saving for a bad day a priority and keeping those savings out of our daily spending by deliberate intent. That little puddle of money is very carefully put away for when we need it and both of use are very clear on what "need it" means. And as others have noted, as soon as you've got a substantial emergency fund, bad days seem to go away which we also experienced.
As I read Hanners, she's got a puddle of money, however acquired, that she can access when she wants but seldom does. Does that make her privileged? Yes, clearly. Does that make her rich in the sense that her mother is rich? Not at all necessary, or at least, we have no indication from the strips that she is.