How does "A hit dog will holler" work as a metaphor?

Background: I, an Australian, once had a co-worker in North Carolina who would often use Southern-US idioms that confused me. I spent an evening panicked about how to handle "This dog will hunt" as feedback on a document before he clarified that meant he was happy with it. Oddly, the phrases always seemed to involve dogs.

So, when a politician from Florida recently used the baffling expression "A hit dog will holler" I wasn't surprised.

Wiktionary explains its figurative meaning:

An offended or defensive response to a statement suggests that the statement applies to the person complaining.

If I understand this correctly, it is like the schoolyard taunt: I'm rubber, you're glue.

But, I understand why the rubber (the insult bounces off me) and glue (the insult sticks to you) idiom works.

What does the yelping of a beaten dog got to do with tu quoque claims?

[Stop Press: A commenter below suggests that I have misunderstood the figurative meaning, even before we get to the literal meaning. Please set me straight on both in an answer.]

Perhaps the fuller version of the folk saying, popularized by the evangelical revivalist Rev. Sam Jones in the early 1890s, would clear up any mystery about the underlying metaphor:

Sam Jones says, throw a stone into a crowd of dogs, and the hit dog will holler.

Quoted from an article in The Ozark Banner-Advertiser (Ozark, Alabama) 31 Aug 1893 (paywalled).

The "authorized" version of the 1885 Sermons and Sayings reproduces Jones's original more faithfully than the 1893 popular press paraphrase:

If I throw a stone into a crowd of dogs, and one runs yelping, you know that is the one that is hit. When you hear one of these fellows on the street yelping at me, you may know he is the dog that is hit.

The underlying metaphor is that an accusation or unflattering description (the 'stone') made about the individuals in a group of less-than entirely respectable people (the 'crowd of dogs') will get a response (the 'holler') from whichever of those people are so accurately described or accused (the 'hit dog').

While the dog may holler, the Rev. Jones did not hesitate to advocate against the throwing of the stone in the first place, with this bit of advice quoted in the El Paso Times (Texas; paywalled), 01 Jan 1893:

"Quit your meanness," is one of Sam Jones' sensible bits of advice to men.

"A hit dog will holler" is not about a beaten dog, but a dog that's been hit, either figuratively or physically - meaning, that the comments made hit home hard and/or applied and made the "hit dog" yelp/holler, or, inferring that the comments made hit the nail on the head and were true as evidenced by the "dog's" reaction to it.

It is similar to Shakespeare's "the lady doth protest too much, methinks" where the overly insistent protests reveals that the opposite may be true about something.

"I'm rubber and you're glue" doesn't apply here as it is merely a schoolyard retort to bullying/name-calling.