But if it happened in Ireland, there was no legal process before the early 1950s, @AInightingale -- so it will have been an informal/private adoption. At least, as I understand it.
It wasn't necessarily a secretive or tightly-controlled thing, either. Absolutely it could be, if it involved illegitimacy, rape, infidelity, a Mother and Baby Home or a Madgalen laundry etc.
But there were also situations where a family member simply 'gave' one of their children to a married sibling who couldn't have children, or had secondary infertility and couldn't have more children, and the child was raised by an aunt or uncle in their family, in loco parentis, with everyone in full knowledge, but no official record. I can think of at least two instances in my own family.
It doesn't seem the most likely possibility in the situation you outline, obviously, but then neither does a married couple giving one of their children up for adoption, as you say, especially if they only had one other child. (The cases where parents gave one of their children to another family member to raise tended to be in larger families, where resources were stretched and it was felt the child adopted out would do better in the other family.) It could be there's more to it, and it might explain why the cousin doesn't want to respond.
(Another complicating factor is that, when adoption was legally formalised in Ireland in 1952, some families took the opportunity to register the adopted child's birth in the name of the adoptive parents, thereby removing any reference to the biological parents from records. But your instance is too early for that. )