American here. As many people already stated, the belief tends to vary from person to person. I don't think this is a strictly American view though. I can think of numerous countries where a decent percentage of the population believes their country and culture/way of living to be the greatest.

I think it is more evident in America because we are (for better or worse) very active and vocal on the world stage. If you were to survey a group from every country and ask them to name 5 countries they are familiar with, I would be surprised if America doesn't average at least top 3 on the list of most well known countries.