We study large $N$ limits of the hyperbolic $O(N)$ linear sigma model ($\text{HLSM}_N$) on the two-dimensional torus $\mathbb T^2$, namely, a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities. After establishing (pathwise) global well-posedness of $\text{HLSM}_N$ and the limiting equation, called the mean-field SdNLW, we first establish global-in-time convergence of $\text{HLSM}_N$ to the mean-field SdNLW with general initial data (under a suitable assumption). In particular, for the local-in-time convergence, we obtain an optimal convergence rate of order $N^{- \frac 12}$ under an additional integrability assumption on initial data. We then show that the invariant Gibbs dynamics for $\text{HLSM}_N$ converges to that for the mean-field SdNLW with a convergence rate of order $N^{- \frac 12}$ on any large time intervals.