What is the sense of electronic configuration of an atom if an electron can be in any superposition of energy eigenfunctions?
The answer that will satisfy you depend on how familiar you are with the mathematics and principles of quantum chemistry. 

So, let's start small. Hydrogen atom. Exact expressions for eigenfunctions for the Hydrogen are known, they are the orbitals we know and love(?). Although the electron can be in any combination of eigenfunctions, it can only stay that way when you are not looking. When you do look, that is, take a measurement, it will collapse to a specific eigenfunction. That eigenfunction could be 1s or it could be 6h, but it turns out that 99.999% of the time, when you measure the energy of a Hydrogen atom, it will display the familiar 1s. That is the sense in which we can say that the electron in a Hydrogen atom is is in orbital 1s, what we really mean is that, typically, for the vast majority of time, the electron of the Hydrogen will be occupying the ground eigenstate.  

Now that is for Hydrogen. For heavier elements, you no longer have orbitals, strictly speaking, you have more complicated eigenstates, obtained by solving the Schrodinger Equation to each atom separately. Currently, there is no analytic solution to the equation when you include a second electron. We approximate the ground state of each atom as though the electron were occupying different hydrogen-like orbitals (hydrogen-like because they differ from the pure hydrogen orbitals in a few parameters that are added). 

Because electrons cannot occupy the same quantum state (considering spin), the way to do that is to assume that each electron is in a spin orbital. The way to distribute the electrons through the infinitely many hydrogen-like orbitals is to distribute them in the way you have learned, following the Pauli diagram. The resulting electron distribution is an approximation to the true ground state, using hydrogen-like orbitals, because the true ground state is not known.  

O hope I made myself understood, but if you need clarification, please ask and I shall edit the answer as much as possible. I have fond memories of the time when I was asking the same questions :)

EDIT 1

You asked why is the probability of finding the electron in 1s orbital highest when we measure its energy? 

The doubt comes because from your understanding these probability comes from writing the initial state as a linear combination of eigenbasis of our choice (in this case the energy eigenfunctions) and the initial state can be any function (normalisable and continuous)

Now, while in principle this is correct, in practice that is not so. When you have a vast sample of hydrogen atoms and you measure the energy of each one of them, the vast majority of them will return the ground state for most experimental conditions. That is because experimental setups that actually manage a high degree of superposition with excited states are very hard to prepare. 

This means that the experimentally expected state for an electron in a Hydrogen atom is 1s for the vast majority of experimental setups. This is the relevant information. The preparation of an arbitrary superimposed state is not trivial, and they do not occur easily in nature.

This leads us to the approximation using hydrogen-like orbitals and the aufbau principle, according to which the ground state (the most experimentally relevant eigenstate) of an atom can be approximated as though each electron occupied one spin orbital around the nucleus, following the Pauli diagram of energies. 

It is not so much a theoretical result as much as a way to make theory and practice agree. There is a way to reach the aufbau principle from first principles, but it involves doing statistical thermodynamics using Fermi Statistics. These calculations would tell you how electrons usually distribute through their available energy levels, and it would become clear that the aufbau principle is pretty sensible, from an experimental standpoint.