In this paper, we investigate the Milstein numerical scheme with step size $η$ for a stochastic differential equation driven by multiplicative Brownian motion. Under some appropriate coefficient conditions, the continuous-time system and its discrete Milstein scheme approximation each possess unique invariant measures, which we denote by $π$ and $π_η$ respectively. We first establish a central limit theorem for the empirical measure $Π_η$, a statistical consistent estimator of $π_η$. Subsequently, we derive both normalized and self-normalized Cramér-type moderate deviations.