[agurk]

I've come across different etymologies for this phrase. The first referenced use I've seen is from 1840 American humorist Seba Smith in her short story The Money Diggers when she wrote: “As it is said, ‘There are more ways than one to skin a cat,’ so are there more ways than one of digging for money”.[0]

This Economist article[1] posits that it is actually slang for sexual intercourse based on the phrase "skin the cat"[2] first recorded in 1837 being a euphemism for it. This interpretation would also fit with the above quote. The same article also points out that:

'And the rather violent act of skinning a cat is no easy thing, says John Youngaitis, a taxidermist in New York. “There is not more than one way to skin a cat.”'

From[0] we can also see that in 1678 English naturalist John Ray said in his “Collection of English Proverbs”: “There are more ways to kill a dog than hanging”.

So it seems it is possible that the contemporary phrase exists to change a less common activity for a more common one. In this case use of the idiom would not convey the sense of doing something unusual.

[0] https://www.bnd.com/living/liv-columns-blogs/answer-man/arti...

[1] https://www.economist.com/prospero/2013/10/09/shooting-skinn...

[2] https://greensdictofslang.com/entry/f3i7zgi