I've got a dog and my dog's name is cat

I've got a dog and my dog's name is Cat
Bet you never heard anything as crazy as that
But not only that, I've got a cat
And my cat's name is Dog, like my dog's name is Cat

I bet you think that's pretty absurd--
But wait 'til you hear about my fish, Bird
And listen to me and you'll hear this
I've got a bird and I call him Fish

With my bird and my fish, my dog and my cat
People all stare when I tip my hat
It's strange as can be when we walk in the park
But it's stranger still when the cat starts to bark