The term you're looking for is "licensing," I think. The copyrights for the series and individual games belong to Square Enix. They have to negotiate deals to get the games on Nintendo and Sony systems, but they still own the games.
As for why the games haven't been remade in the past, I imagine it hasn't been worth the effort until now. Making those old games with Gamecube/Xbox/PS2 graphics would be like throwing money straight down the toilet because I doubt the market is there to justify the development costs and the requisite $60 they'd have to charge, and few people would spend that much for a remake unless it was a package deal.
Doing a $30 port of a GBA title or a $35 remake on the DS probably makes a lot more sense. The graphics requirements aren't as intense, which is forgivable on a handheld system. Obviously, I could be completely wrong. If they have enough success with Final Fantasy IV -- a real test case, since Final Fantasy III was a remake of a game very few people got to play -- I think you'll probably see more remakes.
And if Capcom can port Resident Evil to the DS as well as they did, I see no reason why Final Fantasy VII wouldn't be possible for the system.