The styling “Republic of Ireland” was officially adopted as the “description” by statute in 1948.
This was basically done to allow an in practice name change without amending the constitution. Though more recently they seem to run with the formal name of Eire/Ireland, Republic of Ireland remains an official, common and valid way of refering to the country. Particularly when you need to differentiate with that other Ireland, or the Island they both sit on. Its basically the slightly less official official name. And used contextually as needed.
And the Republic Of styling is probably pulled from the Easter Proclamation which rolls with “The Irish Republic” for the name of the state they done proclaimed.
