We'll take the following definition of a 2-group:
A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$
Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of morphisms, together with maps:
$s,t:\mathsf{G_1}\rightarrow \mathsf{G_0}$ (source and target map)
$id:\mathsf{G_0}\rightarrow \mathsf{G_1}$ (the identity map)
$\circ: \mathsf{G_1}\times_{(s,t)}\mathsf{G_1}\rightarrow \mathsf{G_1}$ (composition map between composable morphisms)
such that the usual diagrams defining a category commute.
One way I see to view a 2-group as a 2-category is to say that a 2-group defined as previously is a monoidal category with the group composition as the tensor product. The delooping category $B\mathsf{G}$ is thus the manner to view a 2-group as a 2-category, am I right?
Maybe the definition you gave is not the most suitable to catch the 2-categorical nature of 2-groups: you are probably interested in this pdf.
Hope it helps!