Lots of tennis being played here, all of which makes me glad I live in a decisive state. Kansas law is first blood. So in the OP's circumstance, if it had happened in Kansas, if nobody showed up, the kid keeps and tags it, if he did show up, as in the OP's story, the old man legally had to tag the deer.

As a hunter, however, it's pretty obvious when someone else has perforated the deer you shoot. Maybe the deer is moving fine and you don't notice a wound before you shoot, but once you touch them on the ground, having an extra set of holes is a pretty good indicator of a "first blood" shot.

I've heard there are states where the law is "last shot," or "killing shot," and I've heard there are many states with no law. I hope in those states, an agreement can be made easily enough between the hunters.

To the tennis match above - if a guy wants to keep hunting and give away his deer, why would he have shot at all? That defies logic to me. It's a different thing to talk about a poacher who uses his buddy's tag instead of his own, that's a far stretch in a conversation about "first blood vs. killing blow" possession rights.