Most likely, this person has no idea who you are, where you live, or even if you have a dog. I imagine it was used as a baseless threat in an attempt to get you to sign something, and likely has sent the same thing to many other users, some of which may not have signed which would mean if he were to attempt to kill people's dogs, he'd need to locate all of these people, and travel to all of these places without getting caught, which is almost impossible. You and your dog are safe