According to Tetsuya Nomura, the concept for having multiple Final Fantasy characters face off in a fighting game originated from Kingdom Hearts. An interview with Nomura in the recently released Kingdom Hearts 358/2 Days Ultimania — a guide book of sorts for the game — sheds some light on how that happened, and how the original idea eventually turned into 358/2 Days.
Where did the idea to use the Organization members for the multiplayer come from?
Nomura: We made a game where characters from various FF games got together to battle, Dissidia, but that was a project we were originally thinking of using for KH. However, deciding it would be a bad idea to make Disney characters fight each other, it never materialized for KH and ended up shifting to FF. Even this time we thought that it would be difficult to bring out Disney characters and have them fight each other in multiplayer, and wondering what to do, we arrived at the idea of using Organization XIII.
These guys were still connected to Disney, but we thought it would be okay because they were original KH characters. Also, with all their attributes and weapons being different, it would be easy to show their individuality, so made a scenario where you could play as Organization XIII in multiplayer. After deciding that, when we were thinking about which time period we should place the scenario in, we thought if we cut out the one year that Roxas was in the Organization it would link well with the multiplayer idea.
If you want to learn more about the development of Square’s three portable Kingdom Hearts games, KHInsider has a translation of two extensive Ultimania interviews with Nomura here and here. The first one has a ton of interesting development talk. We learn that Nomura actually wrote 358/2 Days entirely by himself! (and lived off canned food for two weeks — very John Carmack-ish of him.)