> Does anyone actually claim that Northern Ireland is a country?
It is standard UK government terminology to call it a "constituent country". To just give one random example – https://www.gov.uk/government/statistics/eat-out-to-help-out... – HRMC statistics updated in February this year speak of "Four constituent countries within the UK"
> independence (as opposed to unionism or Irish reunification) is not very popular unlike in Scotland or Wales
I don't think the term "country" for the primary first-level subdivisions of the UK necessarily has anything to do with independence. The term has been used that way for decades, including back in the days when Scottish or Welsh independence were rather fringe positions. It is still used today by many people who oppose the Scottish and Welsh independence movements. So the fact that Northern Ireland independence has very few advocates, is not in itself an argument against Northern Ireland being a "constituent country".
> The UK claims it as a province
Calling it a "province" is controversial. Ireland is traditionally divided into four provinces; Northern Ireland is the six eastern counties of the Province of Ulster, but its three western counties are part of the Republic. Unionist sources often like to identify Northern Ireland with the Province of Ulster; nationalists view this conflation as inaccurate, disrespectful, and politically biased. Due to both the arguments that the term is technically incorrect, and the desire not to inflame the nationalist community, the UK government generally avoids the term "province" for Northern Ireland. Nationalists obviously aren't big fans of the term "country" either, but it is not as strongly associated with the unionist viewpoint as "province" is, so they don't find it as upsetting.
Officially, the UK does not claim "Northern Ireland" to be a "province". I'm not sure if, officially speaking, the UK claims "Northern Ireland" to be anything at all, other than "Northern Ireland", and theirs. Even the original Act of the UK Parliament creating it (Government of Ireland Act 1920) never explicitly said what kind of things the two entities it was attempting to create ("Northern Ireland" and "Southern Ireland") actually were. I believe that was intentional, to try to maximise the chances of the Act being accepted by competing viewpoints (an intention which largely failed to be achieved, but its authors tried.)
Similarly, the Good Friday Agreement never states what "Northern Ireland" actually is. Any answer to that question is likely to upset somebody, so both the UK and Irish governments have decided to avoid all that potential upset by avoiding endorsing any official answer to the question.