The idea of the "American Dream" can be dated back to the book The Epic of America by James Truslow Adams (1931). The author describes the American Dream as
"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 [...] regardless of the fortuitous circumstances of birth or position.
The character of Daisy reunites every one of those traits. She is the "better and richer"; she has also attained a "fullest stature", and to make things even better for her, she was born under "fortuitous circumstances of birth AND position".
These traits make her quite a desirable woman, especially to a man like Jay who visualizes her more as a piece to complete his American dream puzzle than anything more.
Like the idea of an American dream, Daisy, as far as her purpose in life and her personality, can be easily described as "fickle". She weaves herself into the minds of men who describe her and her voice as:
high in a white palace, [she is] the king's daughter, the golden girl
Daisy's aura is that of an unattainable princess. Much like the American dream, she lures, feels "comfortable enough" to be dealt with, appears to be complete, and her voice is tempting, "full of money", musical, and luring. In the end, however, Daisy slips from Jay's hands.
The American Dream is not much different in terms of its nature. It entails the attainment of specific goals having to do with things that signify success. Achieving it is entirely in the eyes of the beholder. However, much like Daisy, you cannot give your everything to it, as it can also slip away from your hands. It can be also quite fickle if it is meant to remain "a dream".
Daisy is also a dream. Jay, who is limited as far as knowing what really should matter in life, quickly acquires the riches, the money, the home, pretends to have an education, and lavishes people in comforts and luxuries...but he still feels that Daisy would complete the equation.
She is his "missing link". As such, he is obsessed with getting her back. Here's the catch, though: he has built such a fantasy of Daisy that he cannot even tolerate thinking that time has passed, that she has moved on, that she has changed, or even the fact that Daisy has even had a child with Tom.
To Jay, Daisy has remained the exact same "girl" that he met prior to going off to war. He expected her to wait for him the way he waited for her. He has made a symbol out of her; a goal more so than a companion for life. The result is easy to guess: he really never gets her, or anything out of her that he had expected. What he gets, instead, is an unstable ex-girlfriend who could care less about him. She is hard to reach. She is a mere illusion in Jay's schema of things.