Well, it's important to both of them, but I don't think its about some sort of objective; verifiable fact of ownership as being about how people _feel_ about something, due to their particular experiences. I guess the thing is that (some) Chinese-Americans have a different relationship to that culture than do the Chinese in China. It has a different significance for the first of those than the second.
I mean, Lord knows, the racial politics around Asian people (meaning, East Asians) in the US is something I find difficult to figure out, because (it seems to me) it's pretty specific to the US experience. Never really thought about it till briefly lived in the US. Racism against black people seems to happen, and have somewhat similar expressions and origins, in many different countries, but there seems to be something very context-specific about the sort of complaints behind this row. Any tensions between white people in Europe and Chinese are likely to be quite different in origin and are going to come out differently over different issues.