The purpose of this expository article is to introduce the notion of a new type of (classical and non-classical) Schottky structures in the discrete subgroups of the projective special linear group over the real numbers of degree $2$. In particular, in this manuscript, we have investigated the classical and non-classical structures of a kind of Schottky group (which we named as semi-Fuchsian Schottky groups) in the hyperbolic plane. In general, a Schottky group is called classical if the Schottky curves used in the Schottky construction are Euclidean circles; on the other hand, it is said to be non-classical if the defining curves are Jordan curves, except the Euclidean circles. In fact, in this article, we initiated the concept of classical semi-Fuchsian Schottky groups of rank $2$ (hence any finite rank) in the upper-half plane model. This study yields new and interesting surfaces in Riemann surface theory, specifically, various types of hyperbolic pairs of pants with infinite area (indeed, we proposed the notion of a pair of full pants) and a hyperbolic torus with one infinite end. After that, we constructed a structure of the rank $2$ semi-Fuchsian Schottky group with non-classical generating sets in the hyperbolic plane by using two suitable M"obius transformations. More precisely, in this paper, we have produced a non-trivial example of the semi-Fuchsian Schottky group of rank $2$ with non-classical generating sets within the hyperbolic plane.