Recently, operator coorbit spaces have been introduced by Dörfler, Luef, McNulty and Skrettingland. A Hilbert-Schmidt operator on $L^2(\mathbb{R}^d)$ belongs to the operator coorbit space $\mathfrak{M}^p$, if and only if its operator short-time Fourier transform belongs to the Bochner $L^p$ space of measurable functions with values in the space of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$. We show that the operator coorbit spaces $\mathfrak{M}^p$ coincide with the coorbit spaces associated with a certain polynomially localized g-frame for the space of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$. Having this characterization available, we can derive approximation theoretic results for operator coorbit spaces.