Mongolian throat singing was not part of the classical repertoire of the Mongolian court. In fact, it is not mentioned in any Mongolian sources before the 20th century. The earliest confirmed mention which clearly describes the type of sound is by a German ethnographer Peter S. Pallas who visited the Minusinsk basin (now in Russia) in September 1772, and wrote:

The Katchin Tatars have the Kalmyk manner of singing in a monotonous sound, from throat, sounded almost like the softly buzzing violin string, which is very pleasant to hear in free air.

Minusinsk neighbors modern-day Tuva, while the Kalmyks originated from the Oirat people of Mongolia (see below) but at this time occupied an area north of the Caspian Sea. Following this, a more detailed report was made the Spanish musician Manuel García in 1847:

Today, among Bashqorts, many people have the amazing ability to produce two quite distinct parts at the same time: a continuous drone (pedal) with sharp melody.

The Bashqorts, today called Bashkirs, are a Muslim Turkic people living in the Ural Mountains. Remarkably, García noted that he also encountered this kind of throat singing among Russian peasants in St. Petersburg.

After García, several Russian ethnographers visited the Tuva area and made cursory notes about their singing techniques, but García's report to the French Academy of Sciences remained the best description for many decades.

In 1921 Mongolia had a Communist revolution. Following this, folk song became a featured part of Mongolian life, as the government wished to show off the music of its new "rulers," the common people. This means that throat singing, which was previously practiced only among pastoralists, was now performed at government-built community centers and promoted to the world. The earliest extant recordings of throat singing were produced by the Soviets in 1927.

So, to answer your question, it's unclear if Mongolian throat singing goes back to Genghis Khan, but it's possible. There is an oral tradition among the Oirat people, who seem to have originated this technique, which traces it back many centuries. The Oirats were not descended from Genghis Khan but took over Mongolia following the collapse of his empire. Carole Pegg documented that the Oirats describe their own singing as being inspired by the sounds of mountain rivers in one of their ancestral homelands, which she was unable to locate precisely, but is roughly on the border between western Mongolia and Xinjiang. If we assume the Kalmyks knew throat singing before they migrated from Oirat territory to the Caspian Sea, this tells us that throat singing already existed in the 16th century. Masahiko Todoriki's article shows that the Oirat lived in southwestern Mongolia from roughly the 15th to 16th centuries.

Citations:

Carole Pegg, "Mongolian Conceptualizations of Overtone Singing (xöömii)" British Journal of Ethnomusicology 1 (1992), pp. 31-54.

Masahiko Todoriki, "Archaic Oirat substratum of the “circa-Altai musical Kulturkreis” in Tuva" Новые исследования Тувы 3 (2017).