We establish the existence and uniqueness of strong solutions, in both the PDE and probabilistic sense, for a broad class of nonlinear stochastic partial differential equations (SPDEs) on a bounded domain $\mathscr{O}\subset \mathbb{R}^d$ ($d=2,3$), driven by multiplicative Gaussian noise. The solutions are global in time for $d=2$. This theory simultaneously covers several physically relevant systems, including stochastic convective Brinkman--Forchheimer equations, stochastic magnetohydrodynamics (MHD), stochastic Bénard convection in porous media, stochastic convective dynamo system, stochastic thermo-magneto-micropolar fluids, and stochastic diffusive tropical climate model, for which previous results only provide analytically weak martingale or pathwise solutions. The proof relies on Galerkin approximation and compactness argument. Up to a suitable stopping time, we derive strong moment bounds and verify a Cauchy property for the approximate solutions, in the absence of any inherent cancellation structure. By applying a Gronwall-type lemma for stochastic processes, we establish the existence and uniqueness of maximal strong pathwise solutions, which are global in two spatial dimensions.