Really? If you forget about modern borders of India and China for a minute, still both Buddha and Shiva are very much tied to Dharmics cultures that originated from the land and people who inhibited the modern day Indian. Culturally, linguistically and racially, they are Dharmics.
Correct, there is no ancient India and ancient China don't include Mount Kailash. Buddhism is most probably Dharmic influenced and I will not go into detail as I am not a Buddhist. Indians like to keep harping about Buddha is from a non existent ancient India like Confucius is from ancient China. I don't know much about Hindu also. I got this Buddha info from a Silk friend of mine. I just watched video of Hindus lynching 2 poor Muslim cow drivers, so I am not too impressed with religions.