People born poor are likely to stay poor because they don't have the same opportunities at birth like wealthy individuals. Young poor kids are forced to find ways to provide for their families at an early age, and this can stop them from focusing on their education. Poor people are also more likely to live in poor urban areas where crime rates are very high.
Unfortunately, history show us that those who grow up poor are likely to remain poor. Sometimes it can be a motivating factor, but more often than not poverty creates challenges that can be insurmountable. Impoverished children often suffer from poor nutrition which affects brain development and educational performance. Also, the culture of poverty does not teach children how to become successful, only how to survive poverty. Their counterparts in families that are more economically stable are learning how to succeed economically from their successful parents.
While it is true that poverty often follow generations, it is not caused by something inherently wrong with certain people. Those in poverty tend to live in areas that also have the worst school conditions and underfunded services compared to those in wealthier areas. It is not wonder they have more chance of staying poor.
The entire concept of the American Dream is at odds with Paul Ryan's perspective. To say "just as likely" is to say you have very little to no chance of being a financially successful person if you are raised in poverty. Obviously this is not the case at all. Poor people have the potential to work very hard with the education given to them, earn scholarships to go to college, and receive sufficient education to become very successful citizens.