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In the Mongol Empire, males could be conscripted into the army at age 15. It therefore follows that men could be qualified for other important duties, such as being Khan, at that age.
As far as I know, there were no Mongol laws against children succeeding at a younger age without having full powers, but they tended not to survive under the severe Mongol infighting. One "Khan," aged 7 or 8, was killed after troops loyal to him were defeated. Genghis Khan himself was orphaned as a teenager, and barely managed to survive the resulting intrigues. The rules of the game may reflect this fact.
To take another example, Sweden had an age of succession of 16 that applied to Queen Christina in the 17th century.
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Among the heirs of Kublai Khan who ruled as both Chinese emperors and Mongol Kakhans, there were several who ascended the throne a lot younger than 15.
After they were overthrown in China by the Ming they remained Mongol Kakhans. One of the most famous of them, Dayan Khan, was put on the throne by the widow of the previous Kakhan when he was 5. She married him when he was a teenager and had many sets of twin boys with him. Dayan Khan had a long and successful reign.
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If you look at the list of the Great Khans you will see it begins like this:
There's a regent every other time so at first glance you might think that the regents were always stepping in for under-age rulers, but that is not the case. It's possible that that game designers were led astray by this different use of "regency". These three regents (two of of them widows) are not holding place due to youth of the successor rather for lack of consensus via election.
Taking each regent/successor in turn:
It's interesting to note that a widow's succession is still a common occurrence in politics.
In addition to the wikipedia citations here, some background comes from Dan Carlin's delightful podcast series (8+ hours!) Hardcore History: Wrath of the Khans