In this paper we present and analyze a fully-mixed formulation for the coupled problem arising in the interaction between a free fluid and a flow in a poroelastic medium. The flows are governed by the Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of stresses and the Beavers-Joseph-Saffman law. We apply dual-mixed formu- lations in both domains, and the symmetry of the Stokes and poroelastic stress tensors are handled by setting the vorticity and structure rotation tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the fluid velocity, structure velocity, and the poroelastic media pressure on the interface as the associated Lagrange multipliers. The existence and uniqueness of a solution are established for continuous and semidiscrete continuous-in-time formulations in both matching and non-matching grids, together with the corresponding stability bounds. In addition, we develop a new multipoint stress-flux mixed finite element method by involving the vertex quadrature rule, which allows for local elimination of the stresses, rotations and Darcy fluxes. Well-posedness and error analysis with corresponding rates of convergences are complemented by several numerical experiments.