Well, looking at some of the practical aspects: If two particle don't interact, then the S.E. is separable and the overall wave function is exactly represented by a product of single-particle solutions. So for an interacting system of particles, the aforementioned product makes for an approximation in the non-interacting limit.
Also, if you neglect the interaction, the single-particle solutions form a complete set. So you can use your single-particle solutions to form a basis for your many-particle system. That's a good idea mathematically if the interaction energy is small, and a good idea intuitively since it tends to be a lot easier to think about stuff in terms of single particles.
Of course, that requires an infinite number of functions, in theory. So you've got to approximate by truncating your description somewhere, and that's where the interesting physics comes into it. :)
Another issue here is that if you're talking about fermions, you have to satisfy the antisymmetry requirement/Pauli principle. That's pretty easy if you're working in a single-particle basis (form a Slater determinant), but easily gets quite difficult if you're not. (c.f. the difficulties of developing DFT methods)
Now for indistinguishable particles the 'identities' are of course just arbitrary labels. But that doesn't bother me. If I'm looking at an atom or molecule, I'm not interested in measuring an individual electron to find out which one it is; I know I can't. What does interest me, though, is finding out the respective states of the different electrons and how they contribute to the overall picture. Because that's how we think about and rationalize electronic structure. Apart from the energy I get from it, the total wave function is in fact rather uninteresting to look at.
