I am born and bred Dublin, our school atlas when growing up showed Ireland to be an Island of the British Isles and most people I know wouldn't bat an eyelid to hear or use the term 'British isles'in the context of the initial question.
Some object for Republican and anti-everything-British sentiment.One I know also never lets you finish a sentance if you say 'the Queen' in it - before she asks which Queen ? Of Englan or the Netherlands
Now calling us British is a totally different argument - But lets not go there.
A humoruos story - well it was at the time - I was in a pottery shop in Malahide (kind of posh Dublin suburb - West-Brit and all that) when I over heard an elderly lady ask the sales person 'Do you have any British or is it all foreign ?) I couldn't resist I just had to interject - 'I'm sorry ! I thought British was foreign....? )
Oh for the look on that ones face ....