I agree with gobluetwo that people do try to weasel around house rules. It's difficult as a host to successfully resolve these situations. If I had been hosting in the same situation, I would have called Airbnb support immediately (there is a number to call, but Airbnb makes it very difficult to find, so many hosts don't know it exists).
I have two rooms for rent in my home and had a situation once where one room was booked, and then I got a request for the other one from people who wanted to bring their dog. I said I'd have to check with the previously booked guests to see what their reaction would be; as a result of that I declined the dog people's booking.
Regarding the cancellation from the host's end, it is definitely possible, but it is detrimental to one's "score" in the way properties are listed, and it also makes hosts ineligible for Superhost status. (Even one cancellation by a host means you're out as a Superhost.) So I'm not surprised that the host didn't want to do that. It all could have been worked out (with no penalty to either guest or host) via Airbnb support, though. They are pretty good about such things, in my experience.