Money management works at all levels. It is just a lot easier when you have more to manage because your tolerance for error is higher, and you can thus be much sloppier and spend less time on it.
What I don't think works so well is teaching everyone to be good at money management. Ignoring the other possible contributors to being very poor (culture, prejudice, whatever), there is undeniably a component due to intelligence. We should thus expect some portion of these very poor to be incapable of managing money, no matter how much they have, and a better policy would be to manage their money for them. This already happens for some mental hospital patients that are out in the real world but need support for medication and housing and food, and if you give them too much unrestricted money instead they'll just buy a PS4 or something else that's shiny and the money will be gone, they'll have to sell their PS4 in a few months to keep paying the rent, and then a while later they'll be evicted due to non-payment. Better to give them a smaller amount of unrestricted money on top of the other aid, but in my experience the same spending patterns emerge (that is, spending it all), very few save, or save beyond what is necessary for a particular shiny toy. There is also a disincentive from getting a job that pays too much income, because their benefits are based on income-received and can be taken away very quickly, but that's another issue. My point is that such individuality-invading money-management could probably be beneficial for more than just the mentally ill, but it will be immediately dismissed by people intelligent enough to manage their own money and who wouldn't be affected by this as nanny-statism.