So first, you might want to see this post from just recently. That was specifically about the Han but it's actually the same situation across the country. There's going to be overlap with my answer here, but I'll repeat some stuff anyway in case people come through and don't click through to that answer.

That seems low.

Wicked low. And also constructed. In fact what the CCP recognises as a single ethnic groups is in many cases multiple ethnic groups with multiple different languages and cultures and customs. The categorisation happened for different reasons with different groups, but one of the most important aspects of it is that it is an offer of status of a sort. By being listed as an ethnic minority, its a form of recognition other groups don't have.

But it's a freaking mess.

To give a sort of example, I work in Tibetoburman linguistics. Tibetan (藏) is one ethnic group. I personally know a dozen people who are listed as Tibetan who come from cultures that vary widely in religion and culture, and of course language. The languages spoken by groups that fall under the Tibetan category are so disparate that it's sort of hard to take seriously the idea of some former pan-Tibetan identity that would have covered all these groups. Yes, there is this idea of Tibetan in the more typical sense, but the modern classification covers basically anyone who was Buddhist of some sort and looked vaguely Tibetan without identifying as one of the other already-extant groups. Zhuang (壯), likewise, is an umbrella term for basically any Tai-speaking peoples of Guangxi, but note that the Dai (傣) are also Tai-speaking. Hui (回) originally just meant they were Muslims, but then in order to get the support of Muslims in the North, Mao offered the benefit of being an official ethnicity.

The number's low, but it used to be a lot lower. Originally there were only 39. In the 60s we were up to 54, and even as recently as 1979 another was added. Don't expect it to change again though. Having official status costs a lot of work and money for the State, and while there are some efforts for recognition of others, it's unlikely to happen.

How did this list of 56 arise?

The sort of ironic thing is that language was often the main factor by which groups were organised. Not always, as the Hui were (are) generally Mandarin speaking and it was their religious practices that set them apart. But for groups like the Zhuang and Yi (彝), language was it. Which I mention as ironic because the guys charged with the task of making the groups simply didn't know enough about the languages to be making that the basis. Yi people speak multiple unintelligible languages, but the organisers were oblivious to the distinctions, or simply didn't care.

Another example are the Gaoshan (高山) people. They're a single nationality that actually includes all Taiwanese aboriginal groups, even though there are actually different language families spoken by the member ethnicities. This is an ethnic group in China that's actually recognised as 13 distinct ethnicities in Taiwan, and is probably actually closer to 16.

You're right that this is an issue of how ethnicity is defined. At the time that this was being done in China it was attempted as a scientific endeavour, but they bit off more than they could chew, and in the end it was simply not practical to continue on. As a result, today you have multiple distinct ethnic groups with distinct cultures and languages grouped together into these 56 group. Even within Han, you have sub-ethnicities, though aside from the Hakka, these internal distinctions have mostly eroded in the face of Han unity.

Unfortunately I can't answer the part about what was taken from Stalin because I don't know anything about the situation in Russia. But if someone else does, I might be able to tell you if the same was done in China for a given action.

Have there ever been any major past or present movements to recognize a particular Han dialect group as a separate official ethnicity or other sorts of quasi-national movements? It seems like the it would be the first step in trying to protect local Han Chinesse languages, and it would be somewhat ironic if the PRC or historical Chinese states rejected this notion, despite promoting multi-culturalism. Also, did Chinese people in Southeast Asia all think of themselves as Chinese, or otherwise why was there conflicts between Hokkien, Teochew, Cantonese etc?