When I saw Kung Fu Panda in the theater, I was impressed with how a mainstream American animated film captured the subtle aspects of my culture. KhoomeiK gets into the details below (I would add Buddhism to Daoism and Confucianism, but that's minor), but the general stoicism expressed by the characters (except Po) is the environment I knew as a child.
Of course Mulan had come out before this, but Mulan was more focused on the narrative and themes around the protagonist. KFP has an ensemble cast beginning-to-end, which allowed for a broader exploration of the philosophy. Each time two characters interact, you get some additional debate that paints a broader picture, leading to the amazing quotes in this article.
At the time, there were even articles about how surprisingly successful it was in China despite being made in the US about Chinese philosophy [1]. You could argue this kicked off what has now become a much larger Chinese film industry.
I'll add from a strictly technical standpoint, the subtle animations for the martial arts are superbly correct. The different animals (Tiger, Crane, etc) are explicit Kung Fu styles. The stances, forms, weapons and training they show are really exact. There's a scene where Shifu corrects young Tai Lung's stance with a stick. The stance is real down to the feet placement in the animation (bow and arrow stance), and in traditional training they do correct you with sticks.
I'll end with one last anecdote: In KFP 2, there's a scene where Tigress says she learned punching "hard style" by hitting an ironwood tree for 20 years. It's written as a joke, but I actually had a Kung Fu instructor who trained that way in real life. Now you train on progressively harder surfaces for punching over a period of 7 years or more, but when he started in the 80's you apparently did it on trees.
[1] https://www.washingtonpost.com/wp-dyn/content/article/2008/0...