Lots on line about the cards but I can't find what department issued them does it say on the card you have?
The article indicates that the Irish Travel Permits (ie those for Irish citizens) were issued by An Garda Siochana (the Irish police). See page 2, penultimate paragraph. If any records of them survive – and I suspect they don’t, because Irish genealogists would probably be aware of them – then I would expect them to be in the National Archives in Dublin.
From other discussions about these documents elsewhere, my understanding is they were in use from 1939 to about 1950.
The UK issued its citizens with National Identity cards during the war, and they were acceptable for travel to and from Ireland, in lieu of a passport, during the wartime period.