In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,χ_q)$, where $χ_q$ is uniformly random Dirichlet character modulo $q$ and $x\in \mathbb{R}$, converges to a random Schwartz distribution $ζ_{\mathrm{rand}}$, which is related to (complex) Gaussian multiplicative chaos. This is the same limiting object that appeared in [34], where the authors proved that the random shifts of the Riemann zeta function on the critical line $ζ(1/2+ix+iωT)$, where $ω\sim \mathrm{Unif} ([0,1])$, converge as $T\to \infty$.