Remember that the Whoniverse is not exactly the same as our universe, just pretty similar. For example, our universe never had a Prime Minister Harriet Jones or President Winters, or a British Rocket Group in the 1950s. On top of that, Whoniverse history sometimes changes—compare the 1986 of '"The Tenth Planet"' to the 1986 of '"Attack of the Cybermen"'. While the writers didn't gratuitously deviate from history in the early days, they got it wrong quite often (e.g., '"The Gunfighters"' had the wrong people present at the OK Corral, and had people killed before they actually died), and the same explanation is generally used in retrospect.
In our universe, Kublai did not have any known rivals named Nogai Khan. There was a Nogai Khan who rose to power shortly after Marco Polo's journey. He ruled the western part of the Left Hand or Blue Horde, which was the western part of the Golden Horde, which was the western part of the Mongol Empire—basically meaning eastern Europe. Although he could have probably ruled the entire Golden Horde, he preferred to retain his freedom as a general and diplomat (especially after marrying into the Byzantine imperial family and conquering Serbia). So, he probably didn't have much interaction with the eastern Empire, and at the time of Marco Polo's journey he was only a minor general in his father's army under Hulegu.
The real-life character who best fits the story is Kaidu. He was never a Khan (Kaidu Khan is the possibly-mythical great*7-grandfather of Kublai and great*6-grandfather of this Kaidu), but he was the de facto ruler of the Chagatai in 1260, and a frequent opponent to Kublai, and was visited by Marco Polo.
However, it's plausible that in the Whoniverse, Kublai did have a rival named Nogai. In fact, Nogai (or Nokhoi), which means "dog", was Genghis's nickname for his best generals and fiercest warriors, and many people had it as either a name or a nickname who could have potentially set themselves up as Khans during the leadership vacuum of the 1260s, with support of the Golden Horde to the west or Ariq Boke to the east.