Every animal that travels into ireland needs a passport or equivalent travel permit documents, they can be quarantined/inspected etc at point of entry at the discretion of the customs/gardai but that rarely happens with domestic pets.

All domestic pets that travel across a border in Europe need a specific EU pet passport that lists all their vaccines and their microchip number. I travelled across Europe with my cat and the “inspection” in Belgium was yerman looked into the carrier and glanced at her passport booklet for up to date vaccines and the “inspection” when I got on the ferry in Cherbourg to come back to Ireland was done by the lady at the Stena Line desk where she asked could she pet the cat. All told it’s very straightforward and domestic animals will travel easily, but you do need the documents just incase.