The meaning of the American Revolution can be, and has been, interpreted in radically different ways by historians. On one side are those who see it as a genuine change in the social order, a transformation as described by David Hackett Fischer and which you've referenced in your question. The "old order of hierarchy and discipline" would mean the European, Old-World social arrangement in which upper-class elites of inherited wealth ruled unconditionally and held power that was mostly arbitrary and unchecked. In this view, the War of Independence supplanted that system with a new one in which the population as a whole became free to exercise power of their own, regardless of inheritance, wealth, and ancestry. It thus represented a true revolution—the overthrow of the existing order.
The opposite view is that the events of 1775 to 1783 resulted not in a change of social order or even in the nature of government but rather a transfer of power from the old elites of Britain to the new ones—Washington, Adams, Jefferson, Franklin, and others—born in the Colonies and establishing a new nation. While those who hold this view may not believe that independence changed the kind of social order already in existence, they generally see at least an intention on the part of the Founders to create a state (the United States) based on principles rather than the murky traditions of the European nations that stretched back centuries to a remote, uncertain past.
Which of these views is "correct"? I would say both are, or, as is usually true in history, the answer lies somewhere in the middle. The Revolution did enable the creation of a new governmental process in the individual states and, by 1789, a central, federal government unique in its division of power among executive, legislative, and judicial branches. There was no genuine precedent in Europe for this arrangement. Central features of British law such as primogeniture were abolished, as was the union of church and state. Both the state and federal governments were based on written Constitutions, unlike the English system or those of the other countries in Europe.
Nevertheless, two main facts prevented the new United States from representing a complete or even fundamental change in the social order/hierarchy. In 1789, when Washington took office under the new Constitutional system, power was held exclusively by white men, and the vote was largely exercised by white men who were property owners. It is true that a by-product of independence was the passing of laws in the Northern states for gradual Emancipation, but slavery remained entrenched in the South. Therefore, in basic ways, the social order, both among white people and across racial lines, did not change.
The second fact is that the principles of the Revolution and those on which the newly independent country was based were not entirely new but were more of an extension of processes that had been occurring gradually in Britain for centuries. England was already a democracy in relative terms, in which the power of the monarchy had been reduced in stages beginning with the Magna Carta in 1215 and proceeding through the changes effected by the Civil War (1642–1651) and the "Glorious Revolution" of 1688. Americans themselves saw the War of Independence as an assertion of their rights as Englishmen, and the ideological struggle within the Colonies was an offshoot of that which already existed in Britain between the progressive Whig Party and the conservative Tories.
So, in conclusion, the events of 1775–1783 were revolutionary in one sense but not in another. Obviously, the secession of the Colonies from Britain resulted in a new nation with different people in charge and a new governmental structure that has maintained itself for 240+ years, but the social hierarchy, the fact that there was still a class structure of rich and poor and white and black within the new country remained the same.
Additionally, we might observe that independence is itself a relative term. American and British people continue today to speak the same language, have much of the same literature, similar music, and essentially the same ideals about the freedom of the individual within society, so the separation of America from Britain in the eighteenth century was not an absolute one, and it isn't so to this day.