What does "whisper thin" mean?

The phrase seems to come more from the verb "to thin":

OED

1 To make thin; to reduce in thickness or depth;

1793 Trans. Soc. Arts (ed. 2) 5 204 The two ends are to be thinned off in form of a wedge.

and from The Freedictionary: we have "thin"(adjective) being "dilute" or "to dilute"

2020 Broth - A thin, clear soup based on stock, to which rice, barley, meat, or vegetables may be added.

2020 Greybeard EL&U "He added turpentine in order to thin the paint."

Thus we see that a soup is thicker than a broth.

"To thin" was then used figuratively in the sense of reduce or dilute.

1787 T. Jefferson Writings (1859) II. 117 Real friends, whose affections are not thinned to cob-web.

Sir A. C. Doyle in "Round the Red Lamp" writes:

"It was then I learned that [...] that rough voice could thin into a whisper when it spoke to a sick child.

Thus we have whisper thin = as thin as a whisper = as weak and diluted (in sound) as a whisper (which is "thin").

So now we have whisper being used figuratively to mean "thin", and "thin" being literal so "the cat was whisper thin" = "the cat was thin thin = "the cat was very thin indeed.