The note for Eidolons indicate they are treated as a summoned creature. I assume this means they are treated as per a Summon Monster spell, with the noted exceptions. Does this mean that the Summoner and the Eidolon share the same initiative result?
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No, not in general.
I'm not aware of any blanket rule that summoned creatures share their summoner's initiative. It's simply a consequence of timing; they appear and act when the spell is complete, which typically means that they always start out sharing your initiative.
Since Eidolons are normally present at the start of a battle, they'd roll their own initiative. (The same would be true if you used summon monster I in advance of combat.) Now, if you summoned one within a combat, in that case they'd appear sharing your init count.
Personally, I almost always have PC entourage characters (familiars, summoned monsters, animal companions, eidolons, etc) act on their associated PC's initiative count, excepting only cases where one of them uses a special initiative action.
The main advantage of this approach is that it tends to make the turn order go much more smoothly, at the cost of slightly disadvantaging characters with high-initiative companions and subsidizing ones with clumsy, low-initiative companions. From my experience, this is a fairly common house rule, but it's still not the official ruling.
The official, by-the-book approach is to give every creature its own separate slot in the initiative order, based on its own separate initiative roll. There is an exception to this rule if the creature is summoned in combat timekeeping (which eidolons usually are not), in which case the creature would start out on its caster's initiative until/unless a special initiative action or leaving and re-entering combat changed it.