Years ago the "Stardancer" (Admiral Cruise Lines) used to allow cruisers with motorhomes and one's dog could live in his or her owner's motorhome and use the provided exercise pens in the bowels of the ship. I assume this service was VERY costly, but I don't know of any ships with this service of late.
QE2 has kennels for pets (dogs,cats) which people can use either for transportation A to B or for just the 'trip'. The kennels are located just near the stack, and for the 'convenience' of dogs, there's even a lampost outside for when they're walked.
QM2 was supposed to have kennels, but they're not in place yet for some reason.