You need a friend recruitment code. The process is:

Step 1) Recruitee purchases and registers the game.

Step 2a) Recruiter logs into Mog Station and generates a recruitment code, which they send to recruitee. (FAQ instructions)
Step 2b) Recruitee logs into Mog Station and enters the recruitment code. (FAQ instructions)

At this point, the recruiter will receive 5 gold chocobo feathers (if one of their first 5 recruits) and Ballroom Etiquette - Improper Greetings (if they don't have it already).

Step 3) Recruitee adds subscription time to their account.

At this point, the recruitee receives a Friendship Circlet, 99 Aetheryte Tickets, Ballroom Etiquette - Improper Greetings, and 10 Silver Chocobo Feathers, while the recruiter will receive a Friendship Circlet and Aetheryte Pendulum and possibly a Draught Chocobo Whistle and additional Gold Chocobo Feathers, depending on how much subscription time was added. (If you're subscribing for a month or two at a time, so your recruiter only gets the first two rewards immediately, they'll still get the others eventually, once you renew that subscription for additional time down the road.)

It sounds like you're past step 1, and just need to make sure both parts of step 2 happen before step 3.