Expressions like your example..
"The Mongols being the Mongols..."
.. basically mean that what you are about to say is typical of the named person or group. Perhaps this is because you have already established this fact about them, or the writer assumes you know because it is widely known or believed.
For example, lets say you were writing about someone named Mary who was known to be quite contrary (that is, inclined to disagree or to do the opposite of what is expected). You might say:
Mary was ordered to go straight home. Of course Mary, being Mary, decided to go for a walk instead.
It is assumed then that the reader knows Mary to be contrary and so going for a walk instead of going home as she was ordered needs no further explanation because it is known to be in her nature to do the opposite of what is asked. If this fact about Mary was not known, you would have to introduce it, and you would instead write:
Mary was ordered to go straight home. Of course Mary, being contrary, decided to go for a walk instead.
I'm no expert on history and know almost nothing about the Mongols, but the history video you link to is speaking about civilisations, and how the definition of the word implies there is some settlement, or permanent land. The Mongols were known to be nomadic and barbarians, meaning they roamed around conquering other civilisations, so quite the opposite of settled or civilised. But they did settle in Yuan China and so are regarded as a civilisation and an exception to the "rule" that would classify them as nomads, I guess. Maybe the video creator is alleging that either the Mongols were known to be contrary, or possibly just the fact that "barbarians" (a term he uses) by nature are rebellious, and therefore prone to the unexpected?