Although it could be all the things as mentioned above, yes, it could be an allergy. I have an allergy to our local milkman's milk - gives me dreadful heartburn, even after just a couple of cups of tea. But, I don't suffer when I have organic milk, nor from many other sources of milk. It's most odd, I know! Occasionally, some cheese will affect me, but by-and-large, cheese seems OK.
It might be something used in cheese manufacture, not the dairy product itself EG rennet(?)
So, try some other sorts of cheese, including organic.