Hmmm, that's odd. It won't show up as equipped, but it should be in your inventory once you open it, and you can equip it from there.
According to SE:
"→ To receive the in-game bonus items that come with FINAL FANTASY XIII (initial batch only), FINAL FANTASY XIV Collector's Edition, or FINAL FANTASY XIV Standard Edition, you will need to register the item codes which comes with the corresponding product. To register an item code, please follow the steps below:
1. Log into the Square Enix Account Management System (https://secure.square-enix.com/account/)
2. Select the "Select Service" option under "Service and Options"
3. Select "FINAL FANTASY XIV," and then "Add a service account"
4. Enter the corresponding item code, and then select "Next"
* The in-game bonus item will apply to the corresponding FINAL FANTASY XIV service account once the registration is complete.
* When you create a character on the corresponding FINAL FANTASY XIV service account, the character will start off with the in-game bonus item in possession. Please note that characters created prior to the registration of the item code(s) will not receive the in-game bonus item, so make sure you register the code prior to the creation process. "
I entered my code on the site before I even loaded the game, so I can't say... but I'd make sure you followed all of these steps exactly, including making sure they are all on the same corresponding service account. You could try deleting all characters and trying again, or even canceling the service option of payment for the characters (though free for trial period), and then adding the option back again now that you have the code input.... that's all I can think of.