Ha, well that explains the mystery. After making a new Square account, linking it to my other slightly-different Square Members account, then going and digging up the logon info on my ancient PlayOnline account to link to the Square Members account which was linked to the Square ID account, which only took about two hours of wandering through the labyrinth of separate yet vital accounts that Square deemed important, I found out that the bonus item for FF XIV was a PS3-version exclusive.

Welp, I had to go through all that account bullshit anyhow before I could play XIV, so it wasn't exactly wasted time. Only a Japanese mind would conceive of such an utterly ass-backwards stack of websites, each with totally unrelated usernames and passwords, and all necessary to get logged in. PlayOnline was retarded enough by itself, with its whole 'we pick your random username for you, good luck remembering it four years from now!' system for logging into FFXI. And yeah, it did need replacing with a less-retarded system. But they didn't replace it. They just merged it in with their new member system, so you still have to be able to lay hands on your old PlayOnline user account info if you want the new system to be aware of your FFXI account. Thank God I'd been smart enough to write down the username on the game box.
And thanks for the code, MuffinMan. Your 360 key didn't unlock any special stuff for FFXIV, but it
did give me these fantastic XBL avatar items:

Which would be fantastic if I actually owned an xbox. But if you want those items, I can PM you the codes for them.