Genghis died in 1227, but that portrait dates to the 14th century. So the artist would have never seen Genghis in person. This is likely a case of an artist applying the cultural standards of one time (in particular associating pudge with wealth) to make a historically important character look good, rather than a genuine attempt at an accurate portrayal.

If I may add a second part to this question, curious to know if the Mongols had any type of sports, or pastimes that they were known to take part in?

Does the same apply to Kublai?