It's been said, but I'm in a ranting mood.
Pokemon Trainer Red is not the same as Manga Red, or Ash from the Anime. The games, ie Red, and Blue, came before either the Anime or Manga came out, and hence any similarities means a reference to the game (not Vice-versa). Nintendo/Game Freak, seeing the popularity of the Anime, essentially "hacked" Red to Yellow Version, with similarities to the Anime (notably, Pikachu walking with you, all starters available, etc). Yellow was the final Kanto game of that gen, and although it's clear Yellow was not a sequel to Red or Blue, the events were different. Yellow, having "taken place" last, is therefore the guideline they followed with Pokemon Trainer Red. In GSC, his team matched that which was possible to obtain yourself in Yellow- Pikachu first (indicating that it was a starter Pokemon, possibly, not to mention it was the highest level). All three Kanto starters, only possible to obtain in Yellow. A Snorlax, two of which can be found in Yellow. Espeon.. Was random. An Eevee can be obtained in Yellow, and i guess they wanted to include a gen-2 Pokemon. Or show that Red was a skilled Trainer who could make his Pokemon happy, or whatever.
Now, in HGSS, Espeon was effectively replaced with a Lapras. This, again, references a possible team for the Player in Yellow, and it also references that Red was indeed present at Silph Co, fighting Team Rocket (remember, the guy gives it as thanks for Red rescuing him?).
From all that, it's clear that GSC were essentially sequels to Yellow, and basically, HGSS aren't sequels to FRLG, but remakes of GSC (there's a difference, as I've shown).
A bit more to prove it references the events of Yellow- look at Green's team. He's got no starter Pokemon, but basically all the Pokemon he'd have, other than the starter. Although in Yellow he had an Eeveelution, that had been determined by the Player's actions, and since the games appear to reference the Player's POSSIBLE actions, then they couldn't assume he'd have a Vaporeon or something.
Hm.. Over two thousand characters. Ouch.