The United States owes a great deal to being a former colony of Britain. The United States speaks English. It places a great deal of responsibility into the hands of a bicameral legislature in a way similar to how Parliament governs. The United States believes in the rule of law and that everyone is born with certain rights. Much of what the colonists complained about during the buildup to the American Revolution was that their basic rights as Englishmen were being taken away.
The United States's foreign policy also owes a great deal to being a former British colony. Just as Britain during its imperial heyday felt compelled to take action all over the world whenever it felt its interests threatened, so does the United States. Britain and the United States have been on the same side in military conflicts for much of the last century. The United States and Britain also enjoy a close trade partnership. All of this is related to the legacy of British rule in the United States.