The War of 1812 contributed to the creation of a uniquely American culture in a number of ways.
For one thing, it made Americans feel more independent and more like a "real" country. It made them feel as if they had stood up to England in a war and come away without losing. This made them feel as if they had truly become a nation that was equal to European countries.
For another, it gave the US more of a sense of nationalism. This was partly out of pride from the war, but it was also partly because the war led to the death of the Federalist Party and the start of the Era of Good Feelings.
Finally, you can see aspects of an American culture coming out of this war since it gave the US such things as our national anthem and the famous line "Don't give up the ship." Both of these served to reinforce the attitude that America was a country of people who fought hard to maintain their independence.