Yes, there is such analog. Let $U$ be a family of subharmonic functions in a region $D$ subject to the following conditions:
a) For every compact $K\subset D$ there is a constant $C(K)$ such that $u(z)\leq C(K)$ for $z\in K$. And
b) $u(z_0)>-C$ for some $z_0\in D$ and some real $C$.
Then there is a subsequence which converges to a subharmonic function. The convergence is in $L^1_{\mathrm{loc}}$ and in $D'$ and in many other senses.
If you remove the condition b), one has to add the possibility that the subsequence converges to $-\infty$ (uniformly on compact subsets). Condition b) can be modified in various ways. For example, you can make $z_0$ depending on $u$ but restricted to some fixed compact $K$.