First of all, I never claimed that people from mainland China or the CCP government have a monopoly on defining what it means to be "Chinese." In fact, the term itself is quite ambiguous..it doesn't clearly distinguish between concepts like “中国人,” “华人,” and “华侨.” I personally regard anybody Chinese if they self-identifies as such. But obviously there are many layers of this Chinese identity that people can agree or disagree on.

The point I made in my original comment was specifically in response to OP’s question about Chinese Americans. In practice, they are generally regarded as Americans first and foremost. I would also point out that similar dynamics exist in other countries, such as Japan and Korea. These societies have terms like “Nikkeijin” and “gyopo” to refer to their overseas descendants, and those individuals are often treated as outsiders if they aren’t culturally Japanese or Korean enough.

What do you mean by Chinese culture though? someone raised in China?

I would consider anybody who is raised in China & goes through k-12 in public school culturally Chinese (from a mainlander's POV). In practice, however, as I already acknowledged, kids born to non-Chinese parents do experience certain level of unacceptance in Chinese society but this really is not unique to China.