This paper presents the mean-square joint state and diffusion coefficient (noise intensity) estimator for linear stochastic systems with unknown noise intensity over linear observations, where unknown parameters are considered Wiener processes. The original problem is reduced to the filtering problem for an extended state vector that incorporates parameters as additional states. Since the noise intensities cannot be observable in the original linear system, the new quadratic vector variable formed by the diagonal of the matrix square of the system state is introduced. The obtained mean-square filter for the extended state vector also serves as the optimal identifier for the unknown parameters. Performance of the designed mean-square state filter and parameter identifier is verified in an illustrative example.