I can't change Oliver the bird's name, he knows it. Oliver the dog probably knows his too. Really, with animals a name is just something we use to focus our energy on one being or another. But, MY energy knows the bird and the dog as Oliver, and changing one or the other would confuse me, maybe them, and maybe the entire pack.
Besides, they are both Oliver. I knew who Oliver the dog was the moment I thought about it. Oliver the parrot is already aware that the spoken word Oliver means him, just as he is sure that SLIM means any dog being, and meow means a cat.
Now either of them would be able to adapt to a name change, but I'm not sure I could.