The American Dream is one of those terms that is intentionally ill-defined. A politcian speaking to a crowd will use the term knowing that just about everyone in the crowd has his own definition, and hoping each will hear the term and think it means what he wants it to mean. GHW Bush used it in almost every speech, but never defined it.
To me, the American Dream is the idea that a person born in poverty can transcend his roots, can get out of poverty by being smart and working hard. We used to call this 'social mobility', and we claimed to have INVENTED it. But these days we do it worse than anyone. The driver of social mobility is good public education, but we've decided we don't want to do that anymore.
Today, as someone said, you'd have to be asleep to believe in the American Dream.
My own American Dream was to have a good career, buy a home, and retire with enough to live on for the rest of my life. Well I'm retired now, house paid for, money in the bank. Medicare. SS every month. MY American Dream came true. But I was lucky. I -wasn't- born poor. Or a minority race. For someone without my own advantages, it would have been a lot harder.