I started wondering this when I tamed the demon dog Verek on a whim. I didn't want to name him Verek, and was trying to think of a good name while finishing BRD with him. But my mind kept going to Paco, which is the name of another demon dog I have, one with the ghostly white skin. Nothing else I tried really fit. It was like the dog said "I'm Paco, I'm just red now, deal with it." So I decided that Paco can materialize in two different forms -- one white and ghostly, and the other red. It fits (kind of) my notion of the demon dogs as cadejos, which traditionally come in two colors/forms... though I don't think of mine as being good and evil.
Paco the Red:

Ghost Paco:

I really like the red skin... I'd never looked at it all that closely before. It's neat when you can suddenly develop an appreciation of something that's been right before your eyes for so long. Red Paco is currently my favorite pet to run around with (despite the fact that the abilities Blizz gave to the dog family are less than exciting).
I'm definitely not going to abandon Ghost Paco just because he's got a new form.

But he may obtain a new, temporary form as a black-and-orange Paco for Hallow's End...

So -- who else re-uses names? Do you ever give a new pet the same name as one you already have? Or use a name you've had before for a pet that you released? Do you think of them as the same character? Or maybe you just like the name...?