Statistically, it's possible. History didn't record how many children Genghis Khan had. He had four sons with one of his wives, and "many" children with other wives, and no one even bothered to record how many daughters he had.

It's been 781 years since he died. If a generation is 20 years, it's been about 39 generations.

If we assume he had 20 children, and each of his children had an average of 2 children, who had an average of 2 children, who had an average of 2 children, he'd have this many descendants: 20 * 2^39 = 109,951,116,277,760.

That's more people than there are on the earth, so obviously, those statistics aren't accurate. But suffice to say: he has a lot of descendants.

It's probably not true that he's the ancestor of everyone in central Asia, though, because people don't breed randomly. There are villages that pretty much only bred within their own village for centuries and centuries; there are religious groups that only bred with each other; there are monastic dynasties that only bred with each other.

Genghis Khan is probably the ancestor of a large percentage of these people. But then again, so are all the soldiers in his Mongol army. If geneological records were perfectly preserved, most people could probably trace their lineage back to him - but also to thousands and thousands of the soldiers in his army. 39 generations is a lot of time for all those lineages to mix.

Idk, but go back 50 generations and everyone is related.
this is because of the multiplier effect stemming from moving backward up your (or any) family tree;
you have 2 parent,
4 grandparents,
8 great-grandparents,
16 great-great-grandparents,
32 great-great-great grandparents…et cetera
and again go back 50 generations and you have over 1 billion in you family. people enough to account for the entire planet 300 years ago. people are in a very real way ALL related, no one is less than 1/50 cousin of one another. A family tree then in fact is diamond shaped if you back enough generations.

and on a side note, the human species is highly inbred.