Isn't it a illusion that poor people bike so much? I mean one of the most efficient indicators of real estate price is travel time to and from a city center.
In Berlin at least there is a strong correlation between income and access to a car. Poor people take public transit and bike much more. They also tend to live close to major roads and suffer from noise and air pollution much more.
75% of the people with an income below 1500€/month don't have a car in Berlin. Only once you reach 3000€/month income (that's roughly the median income) the majority has access to a car.
Poor people can't afford cars. Cramming whole families into cheap apartments of questionable legality is not a great way to live, but you do what you have to.
This sounds highly location-dependent. I'm in rural America, and the poor here, with extremely rare exceptions, own cars. There are people living in dilapidated shacks with SUVs in the driveway.