Before you get on my case about how "racism" in modern usage refers to systemic bias by institutions towards individuals based on their membership in a particular race or ethnicity, I get it. That doesn't mean the word "racist" isn't still completely inadequate when talking about complicated ethnic and tribal tensions, nor the fact that the word stems from the concept of "race", which still retains the idea that people's sociological categorization can be justified using biological principles.
Ergo, "racism" still, in most people's minds and in most online discourse, refers to the "biological basis" of a claim to a certain in-group. And that concept is completely inadequate when talking about ethnic tensions.
Now I'm no expert on the subject, all I've got is my lived in experience, so I'm going to focus on talking about my personal experience of how Americans are completely incapable of understanding how minority works outside of their very race-conscious idea of People of Color. Specifically, I'm going to talk about Chinese-Americans and their inability to understand that "Chinese" is NOT AN ETHNICITY.
Chinese (中國人) is a very modern concept that sprouted up after the concept of nationalism: aka the Nation-State. Prior to the nation-state, there were no "Chinese" persons. Because Chinese refers to "people who belong to the nation-state of China". And the nation-state consists of, at the earliest conception, FIVE major ethnicities:滿(mǎn)、漢(hàn)、蒙(měng)、回(huí)、藏(zàng).
(I will use "Han" instead of 漢 for easier typing in the following texts.)
Note the first: During conception, the first in on the list is a "minority" ethnicity (滿), because THEY WERE THE RULING CLASS, despite the Han being the ethnic majority.
Note the second: 回 is a sinonized Muslim ethnic minority. Their physical features and even names are indistinguishable from their Han neighbors. Of course sinonization is a spectrum (just like gender!), but the majority of 回 people I've met IRL are basically visually indistinguishable from Han and do not practice the Islamic religion.
What I'm trying to get at is two points:
Not all minorities in China were "oppressed". In fact, China had two full dynasties of oppression by minorities of the majority (元、清).
You can't visually tell who is an ethnic minority. There's been too much genetic mixing through out history to draw the line between the majority and minority via bloodline "purity" genetics stuff. Even names aren't always recognizably different.
All of this is complicated by this political fact: In China (PRC), if you do not prove your parents are of a minority ethnicity, you are automatically registered as Han majority. In other words, you literally need to go through extra steps to be recognized as an ethnic minority.
To put it another way: Han majority IS A BIG MISHMASH OF "Not-Minority". THERE IS NO HAN PURITY.
So what is culturally "Han" is basically "the stuff in the middle kingdom that is not an identifier of minority status". Han is not and was never a delineated ethnicity.
( Quick tangent on the origin of the Han as an ethnicity )
TL;DR - There is no genetic basis for Han ethnicity. There isn't even a very clear historical cultural basis for the Han ethnicity, which has been and still is in constant flux.
So, given that very complicated mess of a cultural identity, "Chinese" is, as far as modern conceptualization goes, a reference to citizenship. Aka, do you have legal residence in the People's Republic of China or the Republic of China territories (because the civil war between the Nationalists and the Communists left a giant gaping problem for the modern "Chinese" identity to wrangle with). Granted in the Republic of China, there is a movement to remove China and replace the political entity with Taiwan, so like...whatever, I'll leave that complicated issue to the people in the RoC to figure out.
Meanwhile, in the PRC, which is what people mean when saying "China", what it means to be Chinese (again, a legal citizen of the political entity known as China) on a cultural level is...uh...being fought over. However, continuing the grand tradition of "we have no fucking clue who is part of our in-group so communication is the main standard", the ill-defined social consensus of "being Chinese" (again, here the "Chinese" refers to "person belonging to the political entity known as China" aka 中國人, not Han ethnic majority aka 漢人) is:
Can read and write Chinese (中文). (Note: Written Chinese is grammatically the same no matter the spoken dialect.)
Can speak a dialect of Chinese (中國話). There's like over 200+ of these dialects. The official one is Mandarin, but like...there are definitely grandmas who can't speak Mandarin.
Notice that in the Chinese language, the spoken and written language are conceptually different things. While the written form is unified (more or less), the spoken one is not.
And...that's pretty much it. Any other standard meant to exclude minority ethnic groups will inevitably end up excluding some part of the Han majority due to the fact that the Han majority DON'T HAVE HOMOGENEITY.
So.
With all of that said.
When a Chinese-American complains that their Chinese relatives in PRC don't consider them "real Chinese" because they "can't speak or write Chinese" and that's being "racist"...
...all I've got is STFU. Chinese is not an ethnicity. And if you don't have basic communication skills in the Chinese language, then you need to get out of the discussion of what it means to be Chinese. You are an American. Keep your nose out of other people's domestic problems.
Edit to add:
( More rambling about Chinese written vs spoken and the writing system versus the language and cultural identification with language as a basis )
Edit 2: Following up on the language thing, I felt it's important to acknowledge that there is some attempt to try to equate "central plains" (中原) with China (中國). This is deeply xenophobic, as it excludes not only the officially recognized minorities from their historical influence on Chinese culture in modern PRC, it also stems from a desire to "purify" what is considered "Chinese".
Even before the conception of the nation-state, the political entity known as China has always been a multi-ethnic state. In the modern nation-state conception of the political entity know as China, both RoC and PRC have been founded on the recognition of ethnic minorities as legitimate members of the nation with a political claim to the concept of "Chinese".
Chinese is not about being Han. Chinese is not about being of 華夏 descent. Chinese, in the modern nation-state concept, is about being a citizen of the nation known as China. Anyone who tries to claim otherwise, STFU. (And yes, writing this might also get me into political trouble. It's why I use VPN.)