I agree with music and story. There are even some games that are NOT Final Fantasies that have the Final Fantasy Feel to them. Even if certain mechanics in the game are DRASTICALLY different, a story that makes sense, and an amazing music score with the "flavor" of FFI-IX (and kind of X, for a few songs), accurately describes the Final Fantasy Feel to me. The first Breath of Fire, up until the ending, which was the worst ending I've ever experienced in video game history, had it. Dragon Quest games have it. Golden Sun and GS 2 had it, though GSD didn't. Final Fantasy Tactics, TA, and TA2 had it. FF Mystic Quest, Crystal Chronicles, VIII, XII, XIII, X-2, and XIII-2 didn't have it, while FFX had some of it, but not all of it.

Like has been stated before, though. I'll describe what the feel is like when you can describe what love actually feels like.