Highly depends on where you live, as "poor" is defined relative to the money the median person in your specific environment earns.
Here in Germany, a single person with net earning of less than ~1800€ per month is considered officially poor, while for a family of 4, the limit would be ~4200€.
(and keep in mind that's for Germany, where stuff like health insurance and basic social insurance is already included).
The jobs in this pay region are much more likely to depend on owning a car, partly because of work hours, but also because low income jobs are more likely to be in rural areas.
Whereas higher income jobs offer e.g. flexible work hours, home office and funnily enough, often fewer work hours (as these jobs often are union controlled), making it much easier for the employees to use bikes as their main mode of transportation.
So the mentioned inversion of poor and wealthy regarding the mode of transportation is probably only a thing in developed countries with general higher standards of living.
