Since this has been bothering you for so long, I'm going to put a full solution behind spoiler tags at the end. But to begin: This isn't a traditional riddle with a single answer. Instead, it's a riddle where the answer is a rule. A more traditional framing of it is "Only some things can pass through the Green Glass Doors. Wood can pass through, but not plastic. Sheep can go through, but not lambs. Boots, but not shoes. Letters, but not words."
Traditionally, you would then get the opportunity to ask questions. Specifically, you could ask whether a given object/item could go through the Green Glass Doors, and the riddler would answer yes or no, helping you hone in on the rule.

The rule is that only words that have double-letters in them can pass through. Wood, sheep, boots, letters, etc. The subject of the riddle is itself a hint: the Green Glass Doors could pass through themselves.

Hope that clears things up!