Well, if you are stranded on an Island "alone", paper based money (USD, GBP, etc...) will be of no use as well. However, if you are on an Island with a community, everyone can still reach a consensus of what is a "Store of Value" and a "Medium of Exchange" to use instead of bartering goods & services together.
The intrinsic value comes from the consensus (In Bitcoin case, the network effect). Consensus is actually a network effect. In todays' world, the consensus for Gold as a "Store of Value" is at the Central Banks & governments level. Just as there is a global consensus that the USD is the world reserve currency.
If you know history, you would know that this consensus for Gold & Silver never changed for thousands of years, however, the consensus for the world reserve currency changed many times.