The meaning of "paper" in the context of undergraduate students
In this New Zealand context, a paper is a course and the terms are used interchangeably, usually with one or other being the formal label at any given institution and the other in common use as well. In either case, it is a single class that can be enrolled in individually, with its own final grade, that can be credited towards a degree, diploma, or certificate.

For reference, we can see this guide from Victoria University of Wellington:


  Courses are blocks of work that are taught over one or sometimes two trimesters—they’re often referred to as ‘papers’ by other universities.


or this glossary from the University of Canterbury:


  Courses are 'classes' or 'papers' that involve blocks of lectures usually taught over one semester.


as well as the University of Waikato's Catalogue of Papers, which lists all the courses they offer, or the similar "Search for a paper" page from the University of Otago, and similar items for the other universities.

It does not refer to an exam or to a term paper. "Exam paper" does refer to the physical document given in an exam and the set of questions on it, but never "paper" alone. All of these terms are often used differently elsewhere.