Gilgamesh is not the same character.
It's just a stereotype, like Moogles or Chocobo, if you want pass me the example. Because in some games he is serious, like FFVIII, in other is a comic character, in other something in the middle, like FFVI. There are just iterative elemens in FF games, but there's not a truly connection if not between FFXIII Trilogy and X/X-2 games.
Even the mythology between FFXIII, Type 0 and, apparently, FFXV is similar but it's not the same.
The FFVII world being the same one of FFX it's more like an easter egg than something else. Yes, there were dudes on SE saying they could share the same universe, but that's it, despite that unique easter egg there's nothing else connecting those worlds.
The quote you posted is a hypothesis, since there's nothing hinted at this in the games.