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).
Duolingo isn't a phrasebook - the purpose isn't just to give you a general idea of the meaning of what you are saying, but to give you building blocks that you can take apart and put together yourself to build other sentences that you can use (or recognize) in other contexts. So learning that Is as Éirinn í is in the same ball-park as "She's Irish" isn't very helpful in the long run, and is only going to be a source of confusion when you encounter Is Éireannach í.
Éirinn is one of a handful of historical dative forms that have been retained. While the dative case survives, most nouns do not have a separate dative form anymore, and use the nominative form in the dative case (the initial mutation rules are different in the dative case).
Irish does not have an ablative case.