One of the theories is as Kurt said, that Gilgamesh is a dimension hopping character. And that all Gilgameshs in every FF game are the same one just at different times in his life. There is even a line he has in one of the games that references something from I think his first appearance in V, but I'd have to double check, though it does give more ground to him being the same and that all of the games are just connected by way of alternate dimensions.

As for the XII paradox thing. As I said before I don't believe any of the games exist in the same world, barring sequels/prequels/spin-offs. So I haven't looked at XII to try to find connections with the rest of the series. XII still has all of the ties that you'd come to expect from a FF game though. Airship, chocobos, Espers (reused summon name from past game), classic summons are now airship names, Pharos Lighthouse actually pops back up as Pharos Sirius in FFXIV, which is also a lighthouse. Cloud and Aerith even appear in Tactics as guest characters.

Just because it doesn't have landmarks or common summons, doesn't mean it would be a time paradox. Most FF games doesn't have everything the past games do. Only XIV has done the most to pick from all of the games. The rest of the series make references to past games. The only reference I know of that FFX has is in X-2 with Shinra and that's it. There's no other direct references that they make to past games that isn't already standard for the series. The series is built on the idea that you're starting fresh with a new world, new history and new universe. They aren't going to always be making references to every past game.