I've just finished reading the translation of the short story surrounding Vanille & Fang. It's pretty neat, actually, I suggest you guys check it out. There are others for Sazh/Dajh and Lightning/Snow there, too.

I was swayed slightly towards the idea that they might be lovers when I read this bit from chapter three in the short story
They had promised so long ago. They would be alright as long as they were together. They would never be parted. It was a promise she never wished to break.


But then I read this and felt myself pushed further away from the idea. short story chapter five
Just like when they were children, together against a world of enemies. So long as they were together, they could stay alive. So long as the were together, it didn’t matter where they ran...


So in the end I'm sticking with the "been friends since they were little kids and are effectively (if not actually) family, and have a closer bond than most."

I seriously suggest reading those short stories if you've finished the game. The Vanille/Fang one shows you just how different the cultures are in the two worlds, far more than I thought. I'll probably read the other two tomorrow or something.