Otherwise, the only way this makes sense if every dog is registered and must provide a sample preemptively

That’s exactly what’s happening. I have a friend whose apartment complex has that as a requirement for dog owners to move in. If you leave shit in the grass, they test it, fine you and make you pay for the lab needed to prove it was you. It works pretty well, as she and I went outside one night in the dark with flashlights because I forgot a poop bag and couldn’t find it when I came back with the bag to make sure she wouldn’t catch the fine.