From (1) p. 274
For the shallow water equations, the characteristic speed $\lambda_p(\tilde q(\xi))$ varies monotonically as we move along an integral curve. [...] If $\lambda_p(\tilde q(\xi))$ varies monotonically with $ξ$ along every integral curve, then we say that the $p$th field is genuinely nonlinear.
Existence and uniqueness results need specifications. Are we speaking of classical solutions or weak solutions? Are we speaking of the Cauchy problem (general IVP) or of the Riemann problem (particular IVP)? In the latter case, there is the following result ((2) p. 84):
Assume that for all $k \in 1, ... , p$, the $k$th characteristic field is either genuinely nonlinear or linearly degenerate. Then for all ${\bf u}_L \in \Omega$ there exists
a neighborhood $\vartheta$ of ${\bf u}_L$ in $\Omega$ with the following property: If ${\bf u}_R$ belongs to $\vartheta$, the Riemann problem (6.1) has a weak solution that consists of at most $(p + 1)$ constant states separated by rarefaction waves, admissible shock waves, or contact discontinuities. Moreover, a weak solution of this kind is unique.
Here, "admissible" means satisfying the Lax entropy condition.
More results of this king can be found in (3).
(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002 doi:10.1017/CBO9780511791253
(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996 doi:10.1007/978-1-4612-0713-9
(3) C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer, 2016 doi:10.1007/978-3-662-49451-6
As far as I know there is no uniqueness results (like kruzkhov uniqueness for scalar conservation laws) for the system of conservation laws.\
Depending on the physics of the problem we expect the solution to satisfy some additional conditions. However there is no proofs as such to show that these additional conditions give uniqueness for the $L^{\infty}$ initial data.
Best Answer
Follow the steps in Sec 3.2 of (1). Let's subtract $u$ times the first equation to the second one. After division by $h$, we get the following conservation law for $u$: $$ u_t + (\tfrac12 u^2 + gh)_x = 0 \, . $$ Now, multiply the conservation law for $h$ by $\frac12 u ^2 + gh$, multiply the conservation law for $u$ by $hu$, and add the results. We have the additional conservation law $$ \eta_t + G_x \leq 0 $$ in the week sense, where $\eta = \tfrac12 h u^2 + \tfrac12 g h^2$ is a convex entropy and $G = (\tfrac12 h u^2 + gh^2)\, u$ is the corresponding entropy flux (cf. Eqs. (1.25) and (1.27) of (1)). You may be able to conclude, see (2).
(1) F Bouchut: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws — and Well-Balanced Schemes for Sources, Birkhäuser, 2000. doi:10.1007/b93802
(2) KO Friedrichs, PD Lax: "Systems of conservation equations with a convex extension", Proc Natl Acad Sci U S A 68.8 (1971), 1686-8. doi:10.1073/pnas.68.8.1686