There was/is no rule that said/says a man had/has to be born in Ireland to be able to join an "Irish" regiment or indeed Scotland to join a Scottish regiment etc. The medals you showed at the beginning clearly belonged to someone who was in the Inniskillings, the inscription and the service numbers prove that. Can't comment on the non-military evidence really but it looks to me as if you have your man sewn up. Oh, and at that time before Irish independence in 1922, they were all British anyway. Your man was a volunteer who, generally, could say what regiment he wanted to join and having Irish parents obviously opted for the Skins.
MaxD
