Both...And, Either...Or, and Neither...Nor are the classic examples of
correlative conjunctions.
Either is a disjunctive dual form in English, parallel with both, a
conjunctive dual form, in some uses:
He wants to talk to both Bill and Mike ~ He wants to talk to Bill and Mike.
He wants to talk to either Bill or Mike ~ He wants to talk to Bill or Mike.
Outside of dual reference, however, both and either don't work as
correlatives:
*He wants to talk to both Bill, Mike, and Joe. ~ He wants to talk to Bill,
Mike, and Joe.
*He wants to talk to either Bill, Mike, or Joe. ~ He wants to talk to Bill,
Mike, or Joe.
Both and either can be used pronominally
He wants to talk to both (of them).
He wants to talk to either (one) (of them).
Though only both (as the suppletive dual of the quantifier all) can undergo
Q-Float:
He wants to talk to both/all (of them). ~ He wants to talk to them both/all.
Both/All (of them) want to talk to him. ~ They both/all want to talk to him.
In general, I think it's most useful to consider either as an auxiliary
conjunction; it does some of the work of a conjunction, and appears in an
optional supporting role with other conjunctions in special circumstances.
That's pretty much what all auxiliaries do in English