How do Maxwell's equations uniquely determine ${\bf E}$ and ${\bf B}$ despite no. of equations exceeding no. of unknowns?

Provided that the first two equations hold true at the initial condition, they are redundant for the time evolution, because $$\nabla \cdot \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{c^2} \nabla \cdot \nabla \times \mathbf{B} = 0$$ and hence $$\nabla \cdot \mathbf{E}$$ is constant, with a similar argument for $$\nabla \cdot \mathbf{B}$$. So we actually only have $$6$$ equations determining the time evolution, which is just the right amount.