Uh, Gilgamesh actually originated in FFV where he was a flunkie of X-Death and ended up cast into the Interdimensional Rift (the real Void), and then became an interdimensional traveller bound to it which is how he shows up in other FF games (so far he's confirmed to have appeared in FFI Dawn of Souls, VI Advance, VII Rebirth, VIII, XII, XIII-2, XIV ARR, XV and Type-0 to my knowledge (there were also characters with his name in FFIX and XI but they weren't the same person). He collects 'legendary weapons' but they usually turn out to be poor rip-offs or copies of the real thing, to his chagrin.

Also, there is already a FF fighting game that FFXIV has a connection to, Dissidia (specifically, Dissidia NT which features Y'shtola and Zenos, and the now shutdown mobile game Dissidia Opera Omnia that featured a ton of FFXIV characters, and yes Gil appeared in them too (funnily enough in the earlier Dissidia Duodecium on the PSP he's shown being dragged back into the Dimensional Rift against his will, showing he's now bound to it and has to return to it sooner or later.

Either way, I doubt Namco will want to play ball with SE to bring FFXIV to the SC series, and vice versa with SC characters appearing here in FFXIV - the last few games in that series personally... left a lot to be desired, but that's just my own opinion. Anything's possible I guess though.