Do any cruise ships allow pets such as dogs?

See above. Not counting service animals.

It looks like Cunard is the only line that allows pets, and only on their transatlantic routes. Not sure how they figured out how to accommodate pets while no other line has.

This article has some details though it’s undated.

They have a kennel belowdecks, with a kennel staff that feeds and deals with the byproducts of the other end.

If you have a service dog on many other cruise lines, they set up a toilet area for the dogs on a back service section of the deck, though a friend of mine carries along an indoor pet potty. They leave it set up in the bathroom [apparently they come in assorted sizes, who knew - hers is not too much larger than a basic cat box] and the waste gets dumped in the toilet and flushed, and washed off in the shower, and the shower hosed down thoroughly. If you deal with everything immediately, there is no smell.

White Star Line used to allow pet dogs. There were 12 dogs on the Titanic, for example – only 3 of them survived.

From Carnivals site