In this paper, the author states in the first sentence:
Among the homomorphic images of a semigroup (= a set closed with respect to an associative binary operation) there is at least one group, namely the unit group $I$.
How is this meant, in what sense arises the unit group as a homomorphic image?
If $I$ is the group of invertible elements, if $S - I$ is an ideal, even the Rees factor semigroup introduces a zero element in the image, hence it could not be a group. So how does $I$ arises as a homomorphic image?
In this paper, the unit group is understood as the trivial group with one element, say $G = \{1\}$. Then $G$ is clearly a quotient of $S$. The paper you are referring to is devoted to find the maximal group image of a semigroup, when it exists. Note also that the paper deals with semigroups, which are not necessarily monoids and may have no unit at all.