That is not what is happening in the show. The word Latin refers to his nationality, not his tongue. Like in many shows, English replaces all languages for convenience. For example, nazis speaking English with an accent to each other instead of German.

The Yuan court relied on the semu ren, a highly multi-ethnic official class drawn from outside the borders of China. To put it briefly, the Mongols did not trust the Chinese officials and didn't possess the neccesary bureaucratic skills (or desire for that pencil pushing nonsense) themselves, so they recruited from the greater Eurasian population now within the Mongol empire and beyond, Marco Polo being an example. However, the lingua franca of the court (and the broader Eurasian "silk road" world) was Persian, not Latin. The writers choosing "Latin" as the language Polo speaks (no real indication he did, incidentally) is really a sign of their laziness. Latin was a lingua france among the highly educated in Europe, but there is no real reason to suppose its knowledge had spread outside of the confines of Catholic Christianity.

EDIT: I found my source!

"Latin"(English)

What?

Marco Polo is supposed to be speaking Mongolian in the show (they portray him as having a particular skill at learning foreign languages in the 1st episode). "Latin" refers to his ethnicity as a Venetian. It's possible some Arabian members of Kublai's court spoke an Italian language, such as Venetian (there wasn't a unified Italian language at that point, and Venetian remains a distinct and related language to this day) or the parent language Latin.

I had often wondered how this occurred myself, but I found one part of the answer when I was reading a history of the Mongols for another post on this sub., so I'll contribute what little I can.

When the Mongolian horde invaded Hungary(1241), they sent scouts onward into the west, and one party was caught by Austrian troops. Among them was an Englishman who had been exiled from England, and had gone to the Holy Land. Once there, he had learned several languages and become involved with merchants acting as spies for the Mongols. They transported him to Batu, the later head of the Golden Horde. The Englishman served with the Mongols for some twenty years, as he was taken by them around 1221. He was able to pick up Mongolian through the Muslim interpreters, and eventually became fluent.

This is only one example, but it is a good one I think, because it is a meeting of two cultures from opposite ends of the world. One, from England, was able to travel to the Middle East and learn the languages, before connecting with some from the Far East. If one did it, it's highly likely that more did. Especially with the number of western Europeans making their way to the Holy Land.

There is a book written about this Englishman. It's called The Tartar Khan's Englishman, and it's by Gabriel Ronay.