Bingo: Eircodes were invented for exactly this problem.
Eircodes are unique per delivery point, whereas 35% of addresses were (pre-Eircode) non-unique.
This caused problems for, e.g.:
(1) Delivery people, except the state postal provider (An Post). As you rightly said, An Post kept a note at the local post office (not centralised) of who lives where. For anyone else (e.g. FedEx) it would be literally impossible to tell where to deliver an item, if the address is non-unique.
(2) Ambulances not able to find the correct house.
(3) More complicated provisioning for, e.g., fibre broadband - how do you avoid incorrectly disconnecting a neighbour, if the neighbour has literally the same address as your customer?
Now, an Eircode is sufficient in itself to uniquely identify the delivery point.
To see this in action, go to Google Maps and put in an Eircode, e.g., "D02 AF30 Ireland".
Or go to https://www.eircode.ie/ and put in an Eircode, and the rest of the address will be shown.
But to go the other way (from address to Eircode) literally requires, for non-unique addresses, the person to click on the correct property on a map.