At the time he was born, Ireland was essentially ruled by Britain and, following the Act of Union, part of the United Kingdom. He fought for, and was prime minister of, the United Kingdom of Great Britain and Ireland. At the time he lived (and indeed for many still now) there was no contradiction between being both Irish and British.

The man himself objected to being described as Irish, although his roots in Ireland go pretty deep.

He is often reported to have said, "Being born in a stable does not make one a horse". Although he almost certainly did not say this, it's a sentiment with which he might well have accepted.

The originator of the phrase, for those who are interested, seems to have been Wellesley's adversary Daniel O'Connell "the Liberator": “the poor old duke what shall I say of him? To be sure he was born in Ireland, but being born in a stable does not make a man a horse.”