From everything I've read about the history of the Mongols (and I've read a lot on this topic), I can say that there is nothing mentioned about cats. Perhaps the authors simply didn't pay attention to this, or perhaps there weren't any. There is indirect evidence that suggests there were no cats:
1 - Cats need to be fed, and campaigns often lasted for more than a month. Taking so much food for cats would not be in Temujin's spirit.
2 - Cats are free animals. It would be problematic to keep track of them.
3 - Cats are neither sacred animals of the Mongols nor totemic animals of Temujin.
However, as I already said, I simply haven't read anything about cats. Could you tell me where you got this information?