Let $\Omega$ be the unit ball. Let $f$ be any radially symmetric function and let $V_1 = V_2 = r f\partial_r$, hence smooth vector fields on $\Omega$.

Any and all radial function $u$ solve $V_i \cdot \nabla u = |\nabla u| |V_i|$. Clearly you can arrange for two of them to have the same boundary values while not identically the same.

If you don't like that $V$ vanish at the origin: instead of the unit ball, use the annulus $B_1(0) \setminus B_{1/2}(0)$.

Let $\Omega$ be the unit ball. Let $f$ be any non-positive radially symmetric smooth function and let $V_1 = V_2 = r f\partial_r$, hence smooth vector fields on $\Omega$.

Any and all radially decreasing function $u$ solve $V_i \cdot \nabla u = |\nabla u| |V_i|$. Clearly you can arrange for two of them to have the same boundary values while not identically the same.

If you don't like that $V$ vanish at the origin: instead of the unit ball, use the annulus $B_1(0) \setminus B_{1/2}(0)$.