Origin of "skin" as euphemism for money

Solution 1:

Ultimately it appears to go back to the US Federal Reserve's design of dollar bills.

OED lists your references separately.

P18. colloq. (orig. N. Amer. Business). to have (one's) skin in the game and variants:
to have a stake in the success of something, esp. to have a financial or personal investment in a business; to be closely involved in something.

It is not clear whether the metaphor underlying this phrase is to do with putting oneself at risk (cf. the metonymic uses of skin at Phrases 1c), or with risking one's money (cf. sense 6); both have been suggested.

P1 c. In phrases concerning a person's life and safety, esp. to save one's (own) skin : to protect oneself from injury, punishment, or some other unwanted fate; spec. to save one's own life.

6. fig. U.S. slang. A dollar; = frogskin n.

An investor with "skin in the game" can either risk his money (skin = dollar) or maybe his entire livelihood. Either seems reasonable, although I think I favour the first because the word skin the phrase can be replaced easily with money. It's not easy to go from P1 c to P18.

The golf game has a sense all of its own:

26. N. Amer. Golf. A sum of money offered as a wager or prize to the winner of a given hole, which in case of there being no outright winner may accrue to the following hole. Cf. skins game n. [a game with such sums of money].

Again, skin here refers to money.

Skin as money appears to derive from the colour of dollar bills:

frogskin n.  [with reference to the green colour of the banknotes] slang  (a) U.S. a one-dollar bill; = toadskin n.

† toadskin n. N. Amer. slang. (obs.)  (a) a five-cent stamp;  (b) a banknote.