In general, what is the physics behind seeking a unique solution?
Are there differential equations which, for some mathematical reasons, cannot have unique solutions and hence cannot be used to model any physical phenomena?
(1) There is a correlation between symmetries, conservation laws and boundary conditions (in the language of differential equations) such that if you have enough symmetries (i.e. enough conservation laws) you will obtain a unique solution.
If you have a system with some free parameters left over and there is genuinely no physics left to be put into the model, then you would say that the physics is actually happening in the full space that your equations are working in modulo the free parameters. And there would be a corresponding set of equations that operate only on that quotient space.
The point of integration is killing off dimensions -- if you don't have a boundary condition to fix the integration constant, then the dimension has lived and the integration was pointless.
Maybe here would be a good place to start: Integrable system (Wikipedia).
An example for (2) would be the differential equations that govern what happens when you drop an object without specifying how you drop it.
The physical intuition behind this requirement, or quest, is causality. One feels that given the initial conditions, and given the differential equations thought of as a Law of Nature, then the rest of the solution is caused by the initial conditions and the Law of Nature and so must be unique.
This is a very questionable procedure. Physics does use differential equations for which this so-called « Cauchy problem » is, as we say, ill-posed. Furthermore, there are diverse ways of deciding what the initial conditions should be: some of them are spatial rather than temporal, which undermines the original intuition.
In the original settings you mention, the physical intuition was that you could somehow place electric charges on a surface in a certain pattern and force them to stay there. But since the field was produced by the charges, causality was going on: so the field had to be uniquely defined by the given charge distribution and Poisson's or Laplace's equation.
In Quantum Mechanics, it is okay non-relativistically, but breaks down with the relativistic Schroedinger equ., also called the Klein--Gordon equ., since it is second order, and the physical interpretation has other problems, too.
In General Relativity this breaks down, but it already breaks down for some other equations and some surfaces on which one thinks of imposing the initial conditions.
For an article which is too short, see wikipedia or http://mathworld.wolfram.com/CauchyProblem.html
For a profound, up to date, and 57 page treatment by Prof. Geroch at Chicago, an expert in general relativity, see http://arxiv.org/abs/gr-qc/9602055 or his website.