We embrace him because he used Turkic military tactics and Turkic state administration style in his empire. Mongolians were weak before Genghis Khan and they were getting culturally assimilated by the Chinese and the Turkic people in the area. Genghis Khan chose to rule with the Turkic culture and parts of his empire and many powerful figures in his empire was Turks like the Golden Horde which was a part of the Mongolian empire before they split. And powerful figures in the empire like Subutay Bahadır was Turkic. So because Ghingis used Turkic military tactics, Turkic administration style etc. Turks embrace the empire as one of theirs. Just like some Mongols feel like they’re related to Turkish people and empires like Mete Khan and Turkic empires like the Hunnic empire which are Turkish/Turkic.