In modern usage of the expression “fatten (or sweeten) the kitty” at least, this “kitty” is not a member of the cat family.
The expression is a gambling term, chiefly poker nowadays, and today means to increase the stakes, to add chips to an unopened jack pot.
According to Hoyle, however, “kitty” is “the percentage taken out of the stakes in a game for expenses of any kind.”
In this connection it refers to the “take” of the house, whether the gambling be cards, pool, racing, or other sport.
The source is by no means positive, but we suspect that someone with a fine sense of irony derived this “kitty” as the natural offspring of the “blind tiger” of faro fame.