Character would have to be Boco or Mog, I mean Chocobos and Moogles are Final Fantasy exclusive.
As for the Weapon, there are actually none that appear in every game, just try to name one, I'd say the closest is Excalibur, and that's King Arthur.
Ragnarok maybe, but of course that's a wicked cool spell in Golden Sun, and a ship in FFVII, then I'd say masamune, but not in FFVIII.
Damn FFVIII, screws it all up! HAHAHA
I can't say a weapon, cause there really isn't one that's exclusive to FF and in basically every game, but if I had to choose I'd say Ragnarok.
THE JACKEL