Let us put aside that economic mobility is alive and well and that Obama is wrong factually. Let us also take into account that he is a politician seeking support to do something he wants to do using the coercive hand of government. We'll ignore these and reach for a different point.
The American Dream was never a 'Promise.' It was a result. It was a result of people taking advantage of a system of society that to paraphrase Lincoln is dedicated to the proposition that all men are created equal and at liberty. Liberty and equality gave people control of their own actions and ownership in the result of their own labor. This brought them to relative prosperity compared to what they had experienced in prior societal organizations. This system and its tremendous relative prosperity has existed since England lost control here in the middle of the 1700s.
The term "American Dream" was coined 180 years later in 1931. In its original usage,
[The American Dream is] that dream of a land in which life should be better and richer and fuller for everyone, with opportunity for each according to ability or achievement... It is not a dream of motor cars and high wages merely, but a dream of social order in which each man and each woman shall be able to attain to the fullest stature of which they are innately capable, and be recognized by others for what they are, regardless of the fortuitous circumstances of birth or position. - James Truslow Adams (1878-1949), in "Epic of America, 1931."
180 years of success by the people that worked in this system which was later called the American Dream. The success and prosperity that people created for themselves created the legend that Obama interprets as a high middle class standard of living.
Accurate Reality: The American Dream is not an economic result that can be produced by government programs. It is the freedom to use your own wits within a system were you keep the fruits of your labors.