Did the 13 colonies become more British or more American?

This is an interesting question! Before the American Revolution, most of the colonists, including many of the founding fathers, had a desire to maintain their British heritage. A substantial number of colonists had ties with Great Britain either economically or through family. By the American Revolution, the colonists were the third generation and somewhat removed from the daily experience of their British counterparts. As with many things, time tends to erode the ties connecting them.
After the Declaration of Independence, thoughts about the relationship with the British began to change. Again, even during the American Revolution and the heat of battle, there were some prominent Americans who believed the war could be resolved, and with some tweaks in the agreements with the British, the colonies could remain as part of Great Britain. They may have held a minority opinion, but even some of the early founders held similar views.
The American Revolution ended, and the colonists were left to figure out how to best govern without the British. Here is where it gets interesting! Not wanting to repeat the mistakes the British Monarchy had made with the American colonists and observing unrest in Europe (France in particular), the leaders determined to create a government "unique" in structure but similar in "philosophy." The American form of the Democratic Republic is an amalgamation of Roman, Greek, and British philosophy, but operates in function different from any democracy in the world.
The answer to your question is a little of both. The Americans broke from the British tradition of the monarchy but retained many of the basic structures of British or European governing philosophy.