How do you beat spy island twice?

You couldn't originally. The islands would not let you replay them as the same player, but you could create another username to play them.

By 2011, a restart feature was added, similar to the one used to play Reality TV island over and over. On each island, clicking the small map at the upper right allows you to select "Restart Island." You cannot earn more credits, however, and you never do actually lose the Island Medallion once you win it.