Even though you used the item to call a mount, the actual ability to actually call a mount to ride isn't unlocked until you've cleared a certain point in the main storyline where you have to join one of the Grand Companies.

After enlisting you then have to do FATES for Grand Company seals, which are like a currency which you use to buy things from the Grand Company Quartermaster vendor. This vendor sells an item called 'Chocobo Issuance' for 2000 Grand Company seals.

After purchasing this issuance, you then have to speak to the Grand Company Commander nearby for a quick quest called 'My Little Chocobo' (this quest requires the Issuance you just bought to complete) which at the end of it gives you a chocobo mount that you can name, and which also unlocks the ability to ride mounts, including all other mounts you may have acquired. Confusing and convoluted I know, but that's how SE made it sadly.