I don't quite understand, it sure looks like the first buck had at least 5" on one side, did it not? If not, then the ticket has to be issued. Since your son was licensed for a legal buck and didn't harvest one with the first kill, he was still entitled to take a legal buck, thus the reissue. You did right of course by signing and detaching the first tag, and declaring the error.
Without a reissue of a license, hunters might be incentivized to take an animal that didn't legally qualify, and simply pay a fine for the meat. That doesn't work for the purpose of ensuring only legal take.
Congrats on your son's second deer. He's a very nice buck.