Membership of a regiment does not prove that the soldier in question came from the area 'represented' by the regimental name.
It is likely that the majority of soldiers in an 'Irish' regiment came from Ireland, or an Irish background, but you can't take his being in that regiment as firm evidence of his 'nationality'. You will, I am afraid, have to locate his place of birth, and possibly his parents' place of birth, as that might also affect how he saw his identity.
Remember that, at this stage in history, the whole of Ireland was part of the UK, so there would have been no legal definition of 'Irish' applicable in the military.
Soldiers could move occasionally from regiment to regiment to balance numbers or skills.