Wow, I'm jealous that you got a "New Dog." All I ever had was "Old Dog."
That was always his name, for some reason. I think I gave him up around age 2 or so, not sure. He was just like the story of the Velveteen Rabbit - all the fuzz was rubbed off of his head, one button eye was gone, as well as one jointed leg. He was thrown into a very large toy box in one of our family moves and I found him when I was 16! Incredible.
But, being a sullen teenager, I just threw him back in there and never saw him again. I definitely never got a "New Dog," however.
My brother had "Old Dog." It had originally been a rabbit, but the ears fell off and it became "Dog." He lost it when we were on vacation, and my parents got him "New Dog," but they didn't bond, so I ended up with it.