While the genasi had 2e racial write ups as far back as the Planewalker's Handbook (1996), they certainly have a strong association with the Forgotten Realms, at least for the last decade. There are 3e stats for them in Monsters of Faerûn (2001), the Forgotten Realms Campaign Setting (2001) and Races of Faerûn (2003), and the article in Dragon #293 (2002) also states "Genasi are more common in a Forgotten Realms campaign because of Faerûn's large number of portals to other worlds." And, as has already been noted, their first 4e appearance was in the Forgotten Realms Player's Guide.
Given this, I don't think it is entirely unreasonable to make the claim that the genasi have long been exclusively identified with the Realms. Sure, there are plenty of sources indicating that Genasi can occur outside of the Realms; the same is true of Warforged (which appeared in the Monster Manual III). But I think it is still true that many people would still identify them as being exclusively Eberron or Forgotten Realms creatures.