Being married was not a requirement for achieving the Great American Dream. However, being popular and in demand in society and keeping company with beautiful people who enjoyed socializing and extravagant parties and entertaining with lots of liquor and uninhibited behavior and acquiring lots of money was important.
Tom Buchanan's family history provided him with lots of money - enough to bring "a string of polo ponies" with him when he moved East. However, he had no social connections with the society of New York.
Daisy was a beautiful debutante from a socially prominent family whose connections undoubtedly enlarged their social circle. Her ease in conversing with anyone and making them feel important would have made her a very desirable guest at all the best gatherings.
Tom and Daisy had a marriage of convenience. They understood that they could support each other in ways that allowed them to live a life that was built on enjoying the parties and not worrying overly about how their actions might impact others. Their combined wealth and magnetism allowed them to appear to have reached the American Dream which they were chasing, but they weren't necessarily a happily married couple.