  A graph is called uniquely distinguishing colorable if there is only one
partition of vertices of the graph that forms distinguishing coloring with the
smallest possible colors. In this paper, we study the unique colorability of
the distinguishing coloring of a graph and its applications in computing the
distinguishing chromatic number of disconnected graphs. We introduce two
families of uniquely distinguishing colorable graphs, namely type 1 and type 2,
and show that every disconnected uniquely distinguishing colorable graph is the
union of two isomorphic graphs of type 2. We obtain some results on bipartite
uniquely distinguishing colorable graphs and show that any uniquely
distinguishing $n$-colorable tree with $ n \geq 3$ is a star graph. For a
connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if
$G$ is uniquely distinguishing colorable of type 1. Also, a characterization of
all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) =
\chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we
determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G)
= \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the
family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$.
