He was an American.

I knew him well and soon after the event had from his own lips the whole story.
He came to Monte Carlo with a good deal of money won at draw-poker in a club at Paris and went away richer by some 100,000 francs (about $20,000) than he came.
The catch-line of the song is misleading.

There is no such thing as "breaking the bank at Monte Carlo." This particular player won so fast upon two or three "spins" that the table at which he played had to suspend until it could be replenished by another "bank," perhaps ten minutes in point of time.

There used to be some twenty tables.

Just how one man could play at more than one of them at one time a "foreign correspondent," but only a "foreign correspondent," might explain to the satisfaction of the horse-marines.
I very much doubt whether any player ever won more than 100,000 francs at a single sitting.