Using our previous studies on the nonlinear characteristic Cauchy problem for the Wave Equation in canonical form, we focus on the uniqueness of the generalized solution. We show how uniqueness may be recovered in the homogeneous case by searching for a solution in the space of new tempered generalized functions $\mathcal{G}_{\mathcal{O}_{M}}\left(\mathbb{R}^2\right)$ based on the space of slowly increasing smooth functions in which pointwise characterization exists. In the same way as Biagioni, we can study the sections on the closure of open sets like $[0,T]\times\left[0,\infty\right[$ or $\left[0,\infty\right[$. The uniqueness can be proved in $\mathcal{G}_{\mathcal{O}_M}([0,T]\times\left[0,\infty\right[)$ thanks to an extension of pointwise characterization of elements in $\mathcal{G}_{\mathcal{O}_M}(\left[0,\infty\right[)$.