The reason that Jack tells his sister that it is a medallion, is because it says so in the story. After that, in the Dark woods chapter of the second book, he does mention it three times. A medallion is something that is used for personal usage. It looks like a medal but it larger. The medal is smaller (the size of a coin) and used as a means for achievement (like in sport).
Hope this helps, even though it is quite late.