"She is from Ireland". Can you say "Is as Éirinn di" and "Is as Éirinn í"?
Why would it be the copula here, anyway? I thought that was for noun equivalence, and to me "as Éirinn" is a modifier, not a noun.
In general in Irish, if the copula appears where it doesn't seem to involve noun equivalence, those situations are fixed phrases where there used to be a noun, i.e. they used to be noun equivalence.
In this case in earlier forms of Irish you had:
Is duine as Éirinn dom = A person from Ireland for me
"dom = for me" is often understood as "what applies to me", so the phrase, even earlier, would have been:
Is duine as Éirinn an rud dom = A person from Ireland is the thing for me (which applies to me).
So today we have:
Is (duine) as Éirinn (an rud) dom
This applies also to: Seán is ainm dom = Seán is my name
originally in older forms of Irish:
Is é Seán an ainm is ainm dom = The name which is a name for me is Seán.
Today being: (Is é) Seán (an ainm) is ainm dom
Dative case was an old case used to mark nouns following simple prepositions. A better name would be the "prepositional case". It's needed here because a few words still have dative forms, such as Éire when they follow prepositions.
However, it's worth noting that that only happens in the standard. Among native speech, Éire isn't used, and Éirinn is instead (at least native speech not heavily influenced by the standard).