the mongols just found one dinasty in northern China, the Yuan dinasty, but in game, are Genghis Khan's mongols represented, not Kublai Khan's mongols,
No, Genghis is just the leader. The empire represents Mongolia throughout its history.
Greece has Alexander as its leader - and yet although Alex never ruled Sparta, this is the Greek second city in the game. Because Sparta was an important part of Hellenic civilisation despite never being part of a unified 'Greece'. So it's not "Alexander's Greece" being represented.
England is led by Elizabeth I. While certainly a very capable and politically astute leader Elizabeth ruled a nascent England only a few decades away from civil war, and which controlled a smaller land area than at most major junctures in the country's history - long after it had lost its territories in mainland Europe, before the major British period of colonialism, before union with Scotland, and before the subjugation of Ireland. However its unique ability is 'Sun Never Sets', a phrase coined to describe the vast Victorian-era British Empire, and it provides a naval bonus while Elizabeth's England was not a naval power, although it started down that route during her reign.
America is led by George Washington, and yet the capital city is Washington. Washington governed from Philadelphia and sometimes from New York; although he did move to Washington shortly after its completion at the very end of his term, it did not become the national capital until 1800, three years after his second term ended and a year after his death. His city list includes Chicago, which didn't exist until 1837. Several cities in the American city list - such as San Francisco and New Orleans - belonged to Spain during Washington's tenure as president and, of course, were founded by the Spanish and French respectively, not by the Americans.
Russia is led by Catherine the Great, whose capital is Moscow. During her reign, the Russian capital was St Petersberg.
Most civs, of course, have UUs/UBs that are partially or wholly not contemporary with the leader - possibly most strikingly, the Songhai Mud Pyramid Mosque (like the game's colosseum, a unique structure in reality that's turned into a generic building in the game) is more commonly known by another name - the Tomb of Askia. Fairly obviously, it did not exist during the lifetime of the Songhai leader in the game.
And so on and so forth. The civ isn't defined by its leader figure, who is often anachronistic. Samarkand (spelled Samarqand in Civ V) has always been a Mongol city in Civ games - yet Genghis Khan sacked the city, he didn't settle or rule it. The first Mongol ruler to govern the city was - yes - Tamerlane, founder of the Timurid empire. Which indicates the Mongol civ represents Mongol power at least up to the period of their Timurid successor state. The Timurids themselves gave rise to the Mughal Empire of India.