Stir, stir, stir.

The ability of the Irish singing tradition to adopt and nurture songs from all over the place is something to be proud of. Fortunately songs don't have to carry passports, and noone's entitled to ask you for a passport before you're allowed to sing them either. There's some who might wish they could, I suspect.

It's interesting to know where a song came from, the same way it's interesting to know where a person comes from, but that doesn't mean they shouldn't both be made welcome regardless.

If people fall into the trap of assuming that just because they hear someone from Ireland sing a song that means it must be a song that originated in Ireland, that's just a reflection of what alanabit mentioned, the fact that there's more of a living Irish singing culture than there is in some places.
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Looking through the songs in this thread it occurs to me there are three types of "Irish songs which aren't Irish" - there are those by non-Irish writers, but written as consciously Irish songs. Or maybe "Oirish" songs. These are mostly American, but a fair number from Engand. One that hasn't been mentioned is "A long way to Tipperary". Some of these are pretty dire, but every now and then there's a good one, and that then moves over into a second category.

And that is, songs which have been picked up and get sung in Ireland because people see them as good songs, wherever they come from they get adopted and naturalised.

And the third category is, the songs which are common to the tradition in Ireland and Great Britain, and noone knows where they started.

I'm always pleased that one of the most widely used sets of song books are the four Ossian volumes of "Folksongs and Ballads popular in Ireland", with the remark by the editor in the Introduction "The purpose of this book is to provide you with the song that are actually popular in Ireland". And even though the books don't even set out to be scholarly, they have the notes at the back which indicate the songs that camed from overseas, and where they came from.