This is a complex issue, particularly because people often like to think in terms of an indepedent-particle picture (i.e. the aufbau filling up orbitals), even though the exact many-body wavefunction has strong electron-electron correlations. So let me rephrase your question:
What is the relationship between the KS eigenfunctions and the exact many-body wavefunction?
Mathematically, as you say, the KS eigenfunctions strictly speaking have no physical meaning (as far as we know). However, the KS eigenfunctions do give a useful qualitative (and sometimes quantitative) picture. The reason for this is that the KS eigenfunctions are a pretty good approximation to something in many-body perturbation theory called the quasiparticle wavefunction. The quasiparticle wavefunction is a well-defined physical property of a system that essentially tells you if you add (or remove) an electron with a certain amount of energy, where it will go. For example, see Phys. Rev. B 74, 045102 (2006).
Are there examples of when the KS eigenfunctions don't give a good desription of the quasiparticle wavefunctions? Well there are certainly many situations where the approximations we typically use in DFT (such as the local density approximation) lead to serious problems. However, I don't know of any examples where someone has shown that the exact KS eigenfunctions (i.e. those obtained with the true exchange-correlation functional) don't agree at least qualitatively with the quasiparticle wavefunctions.
As an aside, everything I have said above applies equally well to the Hartree-Fock wavefunctions. In fact, there is a solid mathematical basis for interpreting the HF wavefunctions as an approximation to the quasiparticle wavefunctions. See Chapter 4 of Fetter's Quantum Theory of Many-Particle Systems.
What about the KS eigenvalues? Strictly speaking, in general they do not correspond to ionization energies (or any other physically useful quantity). The one exception is the highest occupied eigenvalue, which is exactly equal to the ionization energy of the system. Janak's theorem tells us that the other eigenvalues are related to the derivative of the energy with respect to the occupancy of that eigenfunction:
See Phys. Rev. B 18, 7165 (1978) and Phys. Rev. B 56, 16021 (1997). It turns out that empirically these eigenvalues are nonetheless pretty good approximations to the true energy levels of the system with some caveats. In particular, the band gaps of solids are systematically underestimated.