This paper investigates the scattering of a three-dimensional elastic shell scatterer embedded in a layered unbounded structure. The background structure comprises a two-layer lossy media separated by an unbounded rough surface. The shell body is filled with an elastic material, the interior of which is vacuum. Given an incident acoustic point source, our aim is to determine the acoustic and elastic wave fields in the space. This problem is known as a fluid-solid interaction problem (FSIP). In this work, the uniqueness of the FSIP solution is proved by using the integral equation method. Based on the decay properties of Greenâ€™s function, the equivalence of boundary integral equation systems and the FSIP is established. Since the system of integral equations containing several integral operators on infinite intervals, we introduce the index theorem of integral operators and analyze the integral operators on infinite intervals to prove the system of integral equations is Fredholm. We then obtain the uniqueness of the system of integral equations and the corresponding existence and uniqueness result of the FSIP.