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