We focus on the numerical analysis of a polygonal discontinuous Galerkin (PolyDG) scheme for the simulation of the exchange of fluid between a deformable saturated poroelastic structure and an adjacent free-flow channel. We specifically address wave phenomena described by the low-frequency Biot model in the poroelastic region and unsteady Stokes flow in the open channel, possibly a macropore, an isolated cavity, or a connected fracture system. The coupling at the interface between the two regions is realized by means of physically consistent transmission conditions expressing conservation of mass, total momentum, and the dependence of the tangential stress on the velocity increment, i.e. Beavers--Joseph--Saffman slip relation. For all model configurations, we introduce the PolydG semi-discrete formulation, which is then coupled with implicit time marching schemes. The spatial discretization hinges on the two-displacement weak form of the poroelasticity system and a the stress formulation of the Stokes equation with weakly-imposed symmetry. This choice facilitate the design of the dG scheme by avoiding the introduction of a Lagrange multiplier and additional penalty terms at the interface. For the proposed method, we present a complete stability analysis and derive a-priori hp-error estimates. We also remark that the PolyDG approximation offers a high-level of flexibility and precision thanks to its arbitrary-order accuracy and capability to support general meshes, which allows a more efficient geometrical representation of complex interfaces. We present a set of numerical results, performed with the Lymph library, on test cases with manufactured solutions and examples of physical interest to validate the error analysis and investigate the performances in practical scenarios.