By definition a group is a set, together with a binary operation $*$ which is associative, has an identity element, and for which every element has an inverse. For example, the integers with the operation $+$ form a group. The identity is 0, the inverse is the opposite. However, the integers with the operation of multiplication don't form a group (what is the identity element? OK, fine, it's 1, but now what is the inverse of 2?). The rational numbers still don't form a group under multiplication, but the rational numbers except for 0 do (the identity is 1, the inverse is the reciprocal).
Back to your question - the underlying set of the multiplicative group is the set of integers coprime to $n$, but the operation is multiplication modulo $n$. It is worth thinking about why this is actually a group operation - there's clearly an identity (1), but why are there inverses?
EDIT: I assume you are looking to understand the answers to your question
Explanation of a mathematical phenomenon?
more thoroughly. Even though the question can be answered without group theory, it's a good motivation to learn group theory since there are many similar questions for which elementary techniques aren't available.
It indeed relates to the multiplicative group of units mod $n$ (in the case that $n$ is prime). The relevant properties of this group for your question is that it has $p - 1$ elements in it, and that the sets you consider in the other question are cosets of a particular subgroup. You will need Lagrange's theorem to get the answer to your earlier question - actually, the proof of Lagrange's theorem is the same as the "elementary" proof of your earlier question.