Because you specifically stated the question as "by the time the Revolution began" the answer is going to lean towards a no. From early colonial times up through the mid-1760's, there was very little to differentiate between citizens of Great Britain and those living in the various colonies around the globe. Most of the culture in America was derived based on populist trends and fads in Britain. Theatre, music, art, fashion, food trends, social commentary, even political expressionism was essentially a greatest hits playlist from the homeland.
There wasn't even the separation of speech that is common today as both the colonies and Britain used rhotic based speech (where the pronunciation includes hard "r"s). Britain wouldn't adjust to the non-rhotic speech they are known for today until much later in an attempt to form a fresh identity away from the United States as it becomes the leading world power in the post WWI era.
The only real differences develop in the late 1760's and early 1770's in the lead up to the Revolutionary War and revolve around taxation, independence, self-governance, etc. Since America didn't exactly have the singular claim to the ideals of not wanting to pay unfair taxes and break away from long standing power structures, that isn't really the development of a unique identity. The American identity, independent of co-development with Great Britain, really does not begin until well after the Revolutionary War and will not expand to the depths that are seen today until after the Civil War.
It is difficult to say that a unique American identity had formed by the time the American Revolution had begun in 1776. The colonists were still British citizens and continued to believe in many British customs. It is important to note that only 40% of the colonists openly identified themselves as patriots at the start of the American Revolution. While the colonists were beginning to view some things differently from the people in Great Britain, these differences had not evolved into an American identity by 1776.
An American identity took a long time to develop. This identity developed as we grew as a country, experienced various events, and developed American ways of doing things. This took many years to occur. Some people would say this identity didn’t fully develop until we became a world power around 1900. Thus, it is fair to say a unique American identity had not formed by the start of the American Revolution in 1776.