Dan-
Irish 'expert' friends say that some English language 'Irish ballads' are descendents of old English ballads, in contrast to the wholly Irish Sean Nos.
Perhaps that's a part of the story of why you think they are Irish too. But I'm sure that music like this has been going back and forth between these islands for hundreds of years, just like Ian C says.
For instance, Thomas Hardy, the novelist, had tune books containing lots of English, Irish and Scots tunes, written out by his ancestors, ordinary country people, before the Great Famine.