I suspect that bit of dialogue with Cid is one of the many moments in the game that didn't really survive the translation from Japanese to English. It doesn't really make any sense with or without Cid's apparent epiphany, but it probably did in the original dialogue. Though wasn't it basically a reference to one of those old black and white movies (Brief Encounter or something like that)? That's how I always saw it. If so it's been translated from English to Japanese and back again and as we've seen from messing with Google Translate that can do some pretty funky stuff
I'd love to see these various moments rewritten now they have a much bigger team behind it.