Op - what you may not realise is that if you say to an English person "your country", they most likely think first of England. For many, "your country" is England and only England.
For others, they may have a wider view and consider themselves to be "British". So after a moment they remember Wales, and bemusedly remember that Scotland is supposed to count, i.e. be included in the "your country" of Britishness, although some of the Scots don't like it.
NI does not register.
It is just not recalled to 'belong', if it is remembered at all.
If we English thought of NI as part of "our country" for even a nano-second, then maybe Brexit would have gone differently.
When NI is recalled at all, it is as an inconvenience, a colony that we should have got rid of ages ago, more trouble than it is worth.
Hence why the trade border is now down the Irish Sea.
Sorry if this causes offence. If as a Northern-Irelander you want to belong to the UK, that is like wanting to be part of a toxic and dysfunctional family with a narcissistic senior member that considers you the black sheep and an embarrassment, when they remember you exist at all.
You would be better off going 'Low Contact' to use the MN phrase, and either becoming and independent state or re-joining the rest of the Island of Ireland.
So, to answer your original question, most people have no idea if NI is part of "our country", what its time zone or currency is, or why they should care.