This is kind of a misleading statement. First off, a dog license doesn't imply that you're a good dog owner or that your dog is a good dog. Animal abuse is sad and common and, if you want to talk eugenics, there are certainly some ugly, smelly dogs. A dog license is more akin to a social security number or birth certificate, which all your little bouncy offspring do generally have to have.
Or, since pets are mostly viewed under the law as possessions and not people, compare the dog license to car registration. If your dog/car is found abandoned on the side of the road, it allows the authorities to contact you and charge you for the cost of impounding him/her/it.
The cars, I'm fine with, but I have to agree with the spirit of this statement (if not the confusing letter). You should have to meet certain requirements to have a child. A simple community college class on parenting would be a-ok with me. The punishment for failing could be as unimposing as being stripped of the tax break a child earns you, and people living below the poverty line could be given the opportunity to re-take it for Earned Income Credit and such. I'm sure I'm not the first person to think of that.
But the same damn thing should happen with pets. Unfortunately, people will always reproduce, but they tend to dump their former best friends in a dark parking lot when they become inconvenient. So any attempt at proactive protection for pets is doomed from the start. At least we have dog licenses, to keep track of some of the fuckers who shouldn't be allowed a dog or a child.