As the title says.
I've already figured out that on the PC version, you just click and drag a chat tab off and it'll create a new chatbox. However on the PS3, this is not possible via analog stick mouse or by connecting a real mouse.
So, how do you create multiple chatboxes on the PS3?
Some functionality and options that the PC version have do not carry over to the PS3... Such as say the Dots for your Inventory and Character Equipment, or the fact your Gil is always on Screen (Bring up Controller menu on PS3 for Gil Display)...
Also, your Gearsets and hotbars are saved locally, so there's that issue too... Yeah, PS3=/=PC... Maybe the PS4 will have these options, maybe not. PS4 at least will have built in Twitch and Cross-Game PSN Chat....