Correct.
On top of that, you need a separate PSN account per FFXIV account as well, so you need to make multiple users with each their own PSN ID if using the same PS3.

I started on the PS3 and then moved to PC myself. My other half and I have PC and PS3 codes on our FFXIV accounts. Between the two of us, we have 4 keys. 2 PC codes, 2 PS3 codes.