This may come from me being Asian (though not Chinese), but I've always seen China as a sort of cultural and economic center - for much of its history, it was the cultural center of Asia, and it was the economic center of the world. Why did everyone from Europe want to get to China? Because it was just so prosperous (relatively speaking). Its cities were some of the largest in the world (even today), and also the most prosperous. Chang'an was the end of the Silk Road and boasted large communities of foreign merchants seeking to make riches from trade; Hangzhou, as another example, boasted some of the worlds' first restaurants and had large water canals to bolster transportation and communication. China's history, in my opinion, isn't really war-centric - more so, it flips between war and peace with the good old dynastic rise and fall.
But, well, that's how I see it. I think the Art of War thing just about falls into the stereotype of Asians knowledgeable about martial arts and all that.
Obviously UAs aren't reflective of the civs' greatest achievements (otherwise Germany would've gotten something else, for instance), but I frankly don't see military achievements as something that a normal person would associate with China - if anything, they'd think of its "ancient" and "traditional" culture, which is what I'm getting at.