The sovereign state that consists of 26 counties on island referes to itself as Ireland.
Variations of The Republic are used for disambiguation but are not used officially.
> Northern Ireland is part of Ireland
The island yes. But not the Republic of the same name.
> and the inhabitants are still called Irish.
A large portion of the inhabitants see themselves as Irish. But there is also a large portion who see themselves as British. These are not necessarily exclusive categories as some of them see themselves as both British and Irish. There are also those who see themselves as Northern Irish.
Full disclaimer: Not Irish or British, just have an unhealthy interest in the situation.
Variations of The Republic are used for disambiguation but are not used officially.
> Northern Ireland is part of Ireland
The island yes. But not the Republic of the same name.
> and the inhabitants are still called Irish.
A large portion of the inhabitants see themselves as Irish. But there is also a large portion who see themselves as British. These are not necessarily exclusive categories as some of them see themselves as both British and Irish. There are also those who see themselves as Northern Irish.
Full disclaimer: Not Irish or British, just have an unhealthy interest in the situation.