Technically, there is such a thing as "a cheese" in British English, although I agree it is probably not what is meant here. It means the whole piece of cheese, probably several kg in weight, as produced in the dairy.
"I eat cheese" would be "Jím sýr" or "Já jím sýr".
While "I have cheese" can also be translated as "Jím sýr", this is not the case vice versa. Of all the languages I have encountered so far, it is only the case in English that "I have x" can also meen "I eat x".