Wikipedia tells me: "Genghis Khan and the following Yuan emperors forbade Islamic practices like Halal butchering, forcing Muslims to comply with Mongol methods of butchering animals, and other restrictive decrees continued. Muslims had to slaughter sheep in secret. Genghis Khan directly called Muslims and Jews "slaves", and demanded that they follow the Mongol method of eating rather than the halal method. Circumcision was also forbidden. Jews were also affected, and forbidden by the Mongols to eat kosher." Why was he so bothered by this?
While we are waiting for experts to chime in, you might be interested by this question and answer train in a previous AMA with the famous Professor /u/JackWeatherford that started with a similar question I had about washing clothes and the professor gives an offhand mention to slaughtering animals by bleeding. In short, regardless of your religion, you are still to obey Mongol laws. There's no religious exemption. However there's no record of anyone being punished as far as Professor Weatherford is aware.
Give the entire AMA a read as well if you're interested.