You can read more about the Summoner class, and their Eidolon class feature, here on the Pathfinder SRD. For the purpose of this discussion, however, the quoted sections should suffice.
Anyway, so the Summoner's Eidolon is an outsider, and the summoner is summoning "an aspect" of it. The details of the outsider and the aspect can be left up to the player, and in a game set on the Prime Material are unlikely to impact the functionality of the class/ability. In a game where the Summoner is exploring the planes, though, this seems like a really interesting thing to focus a section of the game on.
If the summoner meets the actual outsider, face-to-face, what kind of impact could this have on their ability to summon "an aspect" of it?
If the player decides that the outsider is a native of, say, Arborea, can they still summon "an aspect" of it while on Arborea? What if they're on the inner planes? What kinds of items would you use for a Spell Key-type equivalent for summoning an Eidolon? A possession of the actual Outsider?
I don't know, I feel like a Summoner in a Planescape game is a super interesting premise, and I want to know what you guys think? What cool things would you do? I want to talk about the possibilities, here!
The eidolon is sort of this weird mix of extraplanar creature and the summoner's imagination itself. I kinda like the idea of them not being tied to one traditional plane, but some sort of loose planar stuff that the summoner bends and molds with his will. Maybe they draw on specific planes for specific things (angelic eidolons have Celestia-stuff, animal eidolons have Beastlands-stuff), but I'm not sure there's a physical original of the eidolon out there on the planes to visit.