Does the copy of the game bind to the account?

I purchased both the PS3 and the PC version of the game but have been only playing on PS3 due to my PC still being in transit to where I currently reside. Once my PC gets here I'm planning on switching to playing primarily on it while leaving the PS3 version to my little brother. However, I noticed that I can't change the username on the log in screen for the PS3. This is my first foray into subscription based MMORPGs so I am not sure how it works. Does the copy of the game bind to the account? Will I have to purchase a third copy of the game?

Each Square Enix account can have multiple Service Accounts.
Each Service Account requires a platform (PC/PS3) code for each platform you intend to play that account on.
So in your case, until SE sells the code only (if they will), you need to make a new Square Enix/Service Account for your brother and buy a PS3 copy for him.

I see, that is unfortunate. Well, in any case, thank you for the information.

PS3 version (PSN promo code) gets activated towards your SEN. When launching the game first time, you are required to link your SEN to your SQEX, and hence the SQEX ID cannot be changed as it is associated to the SEN.

For different SEN ID, the player will need to have its own copy of FFXIV for PS3, SEN ID and SQEX Account. If the game already installed on the PS3, technically just a PSN promo code is needed to be activated on the SEN for that player.