I think it's fine, better to be named NoName than to have no name at all! As long as the person loves and cares for the animal, I think the name doesn't matter.

I have known a dog named Puppy (which got confusing when Puppy had puppies), and we had a cat named Cat one summer, until he was stolen. He had been dumped out in our neighborhood, and ended up at out house after being beaten up by the male cat across the street. He answered to Cat, so that's what we called him!