Are the names of game tokens capitalized?
Solution 1:
You'd capitalize a specific piece, but not the generic. For example, the Ace of Spades, but ace/king/queen/jack. The Robber, but "a robber token." A knight in chess, but King's Bishop or Queen's Rook.
In regards to your specific example, Mayfair Games capitalizes "The Robber" in its support material for the Catan games. See here.
Solution 2:
There is no reason why these words should be capitalized. I understand they mean something specific in the context where they are used, but they are not names. If a character has a common noun as a name, eg robber, then it should be capitalized.
Solution 3:
I would adhere to the same rules that apply in the real world. In other words, most pieces would not have their names capitalized. However, if a certain piece in the fictional world meets the criteria for a proper noun than I see no reason why not to treat it as a proper noun within the confines of the fiction.
For example, within the fictional world portrayed in the film Monty Python and the Holy Grail there exists a particularly vicious chicken. When discussing the film, this chicken is often referred to as the Vicious Chicken of Bristol because within the context of the film this terrifying creature obviously meets the criteria necessary to be considered a proper noun!
A generic knight piece, on the other hand, would not be capitalized because it is just one knight among many.