As already pointed out the old sailing ships had animals
on board(if you read much about it you will come across many accounts of, "in the worst of the storm a wave washed over the ship and carried away the chickens..."
We keep chickens (not live aboards yet) and it is a lot of trouble to keep the smell and flies down, doable but a lot of work. We also keep ducks (kackie and dark Campbells) which are heartier, lay more eggs, and float(meaning that they wouldn't have to be on board all the time at anchor
or mind too much if thy fell over board), they are also MUCH smarter than chickens and can be easily trained (there is a childrens book called "Ping" about a duck that lives on a boat). IF (and I'm not sure I would) I was going to keep birds aboard I would go with ducks.
The key to keeping any live stock any where is to carefully plan and design their cages and think through how you will work out feeding, watering, and collecting eggs.
P.S. feel free to PM me if you have any questions.