What gives the lie to this is, for example, they rightly say that O'Callaghans come mostly from Munster, but the Aran islands aren't in Munster, or even close. Look at a map. Why would there be a design for a name that isn't local? That said, there may be designs that genuinely correspond to local surnames from the Aran islands, although these may not necessarily be them!

However, some few Irish clans do exist. The Irish Records Office once kept track of the succession of about 20 clan chiefs, although they have stopped doing that. The reason they quit was apparently not that it's unconstitutional to recognise titles of nobility in Ireland, even though it is, but because of an alleged conman pretending to be the McCarthy Mor, i.e. the Great McCarthy, or chief of the McCarthys.

The chief of the O'Callaghans is referred to simply by the title of the "The O'Callaghan" (note that there is not, and cannot be, a definite article in my user name here), and there is a current chief. He lives in Barcelona, Spain! He is also a "Don", i.e. he is both a Spanish and an Irish noble.

There are a dozen or two Irish family tartans (horribly expensive special orders unless you happen to be a Murphy), but most of them don't correspond to recognised clans (I don't think Murphy was ever recognised by the IRO as a clan, for instance), and probably fewer than half the recognised Irish clans have a tartan. Ironically, this resembles how it was in Scotland at one time, including the actual number of recognised clans being quite low, but now it seems like almost every Scot has a clan and a tartan!

The Callaghan tartan is not strictly a clan tartan as would be understood in Scotland, in the sense that it has not been recognised by our chief AFAIK. It was just designed, very recently I might add, by someone called Peter Callaghan, who I understand says that anyone of the name can wear it.