Cheese Dragons are not actually alive. They are enchanted items of food that act almost identically to dragons. They have no need to eat, but they must “live” in humid environments or they will dry out and harden.

Writher and Finninin seem to have an argument over this "dragon": they can't decide which one gets to eat him. Fin's only argument is "He's Not'cho cheese!" Omnom seems to be completely oblivious to their intentions and assumes they're just being nice to him or at least that's how it appears. The Chronicler doubts he actually has the capacity for thought for he is cheese. Cheese is notorious for being the dumbest of all dairy products.