Briefly, the kilt developed in the Scottish Highlands from combining the Irish brat (striped woollen cloak) with the Irish leine (pleated linen tunic), some time after the indigenous Picts (possibly Celts, but if so then Britons, not Gaels) were conquered by the Dal Riada, aka the Scots, a tribe of Gaelic Celts from Ireland.

Tartan was worn by Celts for thousands of years before they ever came to the British Isles, but the patterns (known as setts) had no significance.

Even up to the '45 (the defeat of Scottish rebels in 1745 after the union with England in 1707), few clans had a clan tartan, probably no more than a dozen of them, but were identified mostly by sprigs of different plants worn in their hats.

Tartans with some kind of meaning, including clan tartans, mostly, with these few exceptions, date after highland dress was legalised in 1782, having been banned in 1746, and mechanised woolen mills having been started in the interim, and especially gained momentum after the King gathered the Scottish clan chiefs in London in 1822.

It is at this point that Irish kilts enter the picture, in the 19th century (1800s for the numerically challenged), to largely disappear again by World War Two. They were suggested as Irish national dress, and were plain colours, not tartan, mostly either green or saffron.

As pointed out elsewhere, pipers in military units on both the Irish and British sides of the border still wear the saffron kilt, e.g. in the Royal Irish Regiment (British Army), the Irish Air Corps and the 28th Infantry Division of the Irish National Army. Pipers in the London Irish Rifles (a British Territorial Army unit, i.e. similar to the National Guard in the US) wear solid coloured kilts in something called 'Hodden Grey'.

Irish tartans having specific meanings are a post-war thing, mostly worn by Irish Americans and mostly designed by Scottish and even English woollen mills. They mostly represent Irish counties, although there are two unrelated sets of Irish county tartans designed by different mills, or represent Ireland itself, but there are seven or eight different tartans for Ireland. There are also tartans for each of the four traditional kingdoms, and for maybe a dozen Irish clans and/or names (and no, that does not include any Scottish clans that may be represented in Northern Ireland).

So, all that said, if you have Irish roots and want to wear a kilt, then why not wear a green or saffron one, or a tartan that represents Ireland or the county your people come from? Even if there is a tartan for your Irish clan or name it is likely to be an expensive special order, but you can still afford something relevant without selling the farm.