My reading of it is that people within the have a sense of National identity, which would mean that they would not like to be assumed to belong to another. Perhaps as a Welshman would rather not be thought of as English, a Northern Irishman would not like to be thought of as British. and would rather you didn’t mislead other people about it.
All of which is perfectly understandable, but again, given the limited range of the original question, perhaps not that relevant.
The use of a Union Jack would serve as an interesting starting point for an explanation of the complex and distinct national identities associated with this part of the world, and also to demonstrate the sensitivity with which this issue should be addressed, to avoid causing any unintended offence.