We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an $α$-Hölder continuous drift coefficient, if $α\in (2/3,1)$. The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations, differentiability along subspaces and control theoretic results. We stress that the well-posedness holds despite the Markov semigroup associated to the linear stochastic wave equation is not strong Feller.