Has any nation the right to claim any music as theirs?

All you can claim is that it is music that is currently being made (at that moment) within th current borders of that nation.

If Liza, her dad and all the other members of her family were making the same music in Ireland - perhaps it could be described as being Irish music?

It is just music and it is an international music - which Paddy and co have clearly demonstrated and 'good on em' for that. Most of the people of the world are devided on national grounds. Let's not also do this to music but let the universal language of music unite us.