The people of the islands of Ireland and Britain have intermingled copiously over the centuries. However, in addition to the political divisions on the islands there are distinct cultural differences.
There is a large number of people on the island of Britain whose ancestors (close and distant) travelled from Ireland in search of employment. It is probably for this reason that the citizens of the Irish Republic are allowed to vote in UK elections.
The idea the Republic of Ireland football team temporarily become Brits during a tournament (as iain has stated above) probably only applies to deluded BBC commentators. The English team would be reviled by all supporters in the Republic (and a fair few in Northern Ireland, Scotland and Wales) even if they were playing the Pol Pot XI.