Oh dear, its seems you are struggling to understand how a referendum works.
Never mind
The result of a vote or a referendum does not change peoples' votes.
It does not change their preferred option.
It does not change their preferred policies.
It does not affect who they prefer to be PM, or local MP, etc.
NI voted in the referendum to remain in the EU.
The result of the overall UK referendum cannot change that.
The government can (and did) decide to ignore the regional differences. Because the government is ignoring the regional differences, it is going against those (regional) peoples' wishes, and in agreement with the wishes of the people from other regions.
What the question asked and did not ask
This means there can be no explicit mandate from the terms of the question for, say, Scotland or Northern Ireland to remain. There is no explicit mandate either for Gibraltar to leave.
There is nothing, at least explicitly, on whether Britain should remain part of the EU single market or customs union
the fact remains that these consequences are not expressly stated. And without them being spelled out, there is scope for doubt and dispute.
In other words, there was a mandate for the ultimate objective but not for how and when that would be achieved. But quite quickly a number of other things were said to be mandated by the referendum.
https://www.ft.com/content/b56b2b36-1835-37c6-8152-b175cf077ae8
The government applied the criteria for the whole of the UK to leave, despite obvious regional differences (Scotland, NI and Gibraltar.) It did not have to apply that interpretation.
The referendum did not refer to the type of Brexit,yet the government chose to apply it to exiting Customs Union and Single Market. (Brexit means Brexit) It was not mandated to apply that interpretation.
Quite obviously the government have picked and chosen the interpretations of the referendum question to suit their political ends.
Therefore, it is absolutely essential that parliament has a meaningful vote on the type of Brexit. (Even if they have been denied a vote on whether the referendum applies to the whole of UK and Gibraltar.)
Did they have a meaningful vote on when the Article 50 was triggered? So the government has forged ahead with their own interpretations of the referendum question.