I can't really answer your question in the way you would expect. I'll elaborate.

I'll preface this by saying that not all colonists supported the revolution; remember, most of these people still had friends and families in Britain. They also didn't particularly identify as "American", I believe they'd most likely identify themselves with whichever colony they belonged to (ie in Virginia, you might identify as a Virginian). The word "American" at this point in time (1600s to 1700s) would probably still refer to a Native American. To add to this, Americans really didn't have much of a sense of national unity until the Civil War; I'm sure you've heard the classic "before the civil war, people said the United States are, and after it they would say that the United States is".

However, they didn't need a collective national identity to go against the British. While they might not have considered themselves American in the sense that we would use today, they certainly wouldn't consider themselves British. These were rebels, after all, people we (in america) would probably see today as 'terrorists' or insurgents.

If you take all this into account, your answers itself. They stopped identifying as British for the sole purpose of rebelling against the British; and that is probably where their identity comes from.

edited for grammar