The Irish are not and have never been British.
Not British in terms of ethnicity, in terms of living on the eastern island of 'The British Isles', but politically, as far as citizenship went, the Irish were considered His/Her Majesty's subjects from 1801 on, and consistent and strenuous efforts were made to convince the Irish that they were joined in an Imperial brotherhood with their neighbours and betters on the other side of the Irish Sea.
This was whether they liked it or not. Those who liked it were ready to commit treason to defend their status (the Ulster Unionists in the run up to WW1 and their allies in the British Army).
One of my grandfathers, who later joined the (old) IRA was obliged to recite the following comforting rhyme in his primary school in the rural south east of Ireland, in which no Irish was taught:
"I thank the goodness and the grace
Which on my youth has smiled,
To make me in these Christian days
A happy English child"
And another that finished up with:
"Britain thou art my home, my rest,
My own land, I love thee best."
Grandad visited London once 
.
He expressed the hope to me when I was heading to secondary school that I would learn about the geography and history of Ireland, since that opportunity has been denied to him in schools where the reading materials intentionally avoided any mention of Ireland, its landscape, traditions, history, music, or culture. If Ireland was mentioned at all it was emphasised that Ireland belonged in the British Imperial world which was glorified as the epitome of civilisation, with national differences minimised and even linguistic ties underlined. Books that were used in Irish schools for children up to age 14 were very popular throughout the Empire because of this pro-Imperial slant.
I highly recommend the book "Benefits Bestowed?: Education and British Imperialism", edited by J. A. Mangan, for a discussion of British Imperialism at work in the educational field.