Ah... guys? The coins aren't just generic discs. They have designs specific to the site they were created in, likely in a pattern that would be difficult to fake and that only a few people know how to make well. That means most people won't be able to make coins, regardless of whether its wood or metal. In real life wooden coins would be easier to counterfeit, but elven wooden coins would probably be made by convincing trees to produce wood coins (however they do that for their other wooden products), likely in patterns that would be impossible to imitate through carving.
Coins are not a fiat currency though. Having specific designs means you can authenticate the weight and composition of a coin without a scale, but for a flegling nation, declaring a currency is not enough to make people trust it. People would much rather store their wealth in tangable stuff than paper money, if the authority printing the money is in danger of being overrun by invaders.
The trees that elves grow stuff from are not rare. In every town there is at least one elf who's proicent at growing stuff from trees, if they aren't all. Wood, even grown wood, is not a scarse resource in eleven society. It does have intrinsic value, to make anyone trust it in the event of the demise of the issueing authority.