Great replies all. And after reading them, I see that I should clarify some points a bit - might have went a little 'off' in the original post. I'll stay away from the bullet points that I so love, however...
First off, I like cats and dogs. Both can be great companions - or not - depending on the individual pet and owner. We treat our dog as a less-capable child that does not require a college fund; yet she is part of the family and not a plaything or toy or even a 'pet.' A cat in our household would be treated similarly.
When I say that the One Ring is a cat, I mean that the Ring exhibits the stereotypical cat behavior and traits. Why? Well, first, the One Ring is not Sauron. You can't have two things be the same thing, if that makes any sense. Sauron, in making the Ring, did not clone himself, and even if he were able to do so (NOT!), the clone would be his creation and not the original. There would be inherent limitations in the secondary, and so the Ring could not be as Sauron. But I digress.
The Ring was not Sauron's child. Surely some of
me is within my children, yet my children exhibit behavior that is not Ring-like (besides the obvious, yet the one daughter tends to tempt others into evil deeds

). They are autonomous; well, they will be one day I hope. The One Ring was more parasitic in nature in that it requires a host to truly be active. My kids are separate from me and can exist without me. We share many things in common, yet my children may grow up not liking Tolkien! The Ring had no such choice. It was like Sauron, and shared much of his desires.
The One Ring was not a new creation unto itself. Again, it could not reproduce, could not do much without a 'helping hand.'
So my thoughts ran thus: The Ring was not Sauron, was not his child, was not a separate autonomous individual, and so the best that I could imagine was that the Ring was like a pet of sorts - semi-autonomous and having many traits of its owner, yet also having its own special individualism.
Our pet, having been with us since puppydom, has taken on many of our family's characteristics while also adding her personality and uniqueness to the mix. It's not been that we've trained her or that she's trained us; it's been a mutual growth experience. Still the dog is more dependent on us than we on her. She (boy is she becoming famous), theoretically, could live on her own, though I'm guessing that she's really too domesticated to truly do so. She then requires an owner or a host.
The owner or host could be someone else - not me. The current host would acquire the benefits of her companionship. She protects us, she plays with us and is a great companion/member of the family - the new owner would get all of this (and for $10, she's yours!

).
But, in reality, this wouldn't work with my pet. Not only is she nuts because her genetics, we've treated her so much like family that for her to change owners would break her heart (we've left her at the parents' house for an extended weekend while we were away and it was bad). So, though the Ring might be a pet, it was not as my dog.
In the past, as mentioned, we owned a cat. Again, the cat was as much a family member as is the dog, yet the cat would catch and kill rabbits, mice and other assorted small animals, and so I would think that the cat could live on its own. Also, as I saw the same cat eat at the neighbors and also play with the same, I'm guessing that the cat could survive if we'd left town suddenly. Surely it may miss us, and we him, but to me the pet's chances of surviving and even thriving a host change seem to be greater if the pet is a cat and not a dog.
And so the One Ring is a cat.
Hopefully that was a little more clear.