My personal rule of thumb would be that if families are allowed aboard, so are non-contained pets. Fish or any aquarium/terrarium pets are to my thinking different than dogs, cats, or anything that needs to roam around. This may be oversimplifying, but if you can't have your kid aboard, I don't think you should have your puppy aboard either.
One might also argue that the plants (Beauregard
) in TOS were part of the ship's botanical research and not strictly personal pets.
What are real-world military regulations like?