As far as I know, indiscernibility is used in two ways in model theory. One, as you say, is to obtain many non-isomorphic models. This is for sure the classical use of indiscernibility.
Another, more modern one, is to have access to tools such as Ramsey's theorem and its uncountable version, the Erdős-Rado theorem. This is useful in some formulations of stability theory or (more recently, as in the work of Byunghan Kim) of simplicity. The point is that the notions of forking and dividing are cleaner to formulate in the presence of sufficiently indiscernible sequences. (So one typically works in large saturated structures in this context.) There are several modern references for stability, etc, where the use of indiscernibility is apparent, see for example Frank Wagner's "Simple theories", Mathematics and its applications, Kluwer Academic Publishers, 2000.
A third use of indiscernibility is fairly common in set theory, where it is the most common approach to defining the large cardinal notions known as sharps. A good reference for this use is Kanamori's "The higher infinite".