It comes down to the situation and context. If someone is complaining that a company is screwing over consumers and doesn't like that, they are allowed to do so. It's not the biggest problem in the world, but to them it's personal enough that they want to complain about it to make changes. There are things that might be bigger, but it doesn't make them less right, or less personal. Now if they are trying to claim that it's a bigger problem than other things, then one can say that they are privileged enough to be able to voice that type of complaint, instead of living in fear of dying, and not being able to speak out. I haven't seen this myself personally.
The harder part comes to those complaining about how bad they have it, or complaining about something in life particular to them. It's fine if they want to complain about work and other life related stuff. When I was a Life Guard, I would vent steam by talking about the stuff I had to deal with. If I'm just venting about the stuff that I dealt with, and that wasn't pleasant to me, that's fine. If I decided today to go talk about how my life sucks, and nobody can imagine my pain, then people can start to make that argument. I have a steady job, don't have to really worry about being discriminated against, etc. where as there are people in the world that have more to worry about than those things, so people will call me out on my bullshit. If a person is having a bad time period, and talks about it, that's fine. The problem comes when you start trying to make people think you have the worst, when there are clearly others that do. Note, I am talking about when small, mundane things have not been working out. If someone is complaining about someone close to them who died, if they got raped, etc. then don't use the privilege to complain card on them.