My understanding is the following: You can only get exact analytical solution for the Schrödinger equation for a) a single electron and b) a single proton - nucleus. That is, for the hydrogen atom with a single electron. The solution for this atom exactly describes the wavefuntion for the electron in any of the orbitals (the base one, or a excited one).
When you have more than an electron, and/or more than a nucleus, then you cannot get exact analytical solutions for the Schrödinger equation.
(I am not sure, if you have a multiproton nucleus with a single electron, e.g. He+, if you can get exact solutions; I believe so).
As for atoms with multiple electrons, you can consider that the efect of other electrons shields the charge in the nucleus, and then you can approximate the orbitals in this case to the single electron case.
When you have multiple atoms forming a molecule, and there are shared electrons positioned around two or more of their nuclei, you cannot get an exact analytical solution - even if you still had a single electron, as in H2+. But it was found that by adding or substracting the atomic orbitals, you get a reasonable approximation for the molecular orbitals that form covalent bonds. Eg. By combining the s orbitals of two H atoms, you get a close approximate solution of the molecular orbital of the H2 molecule. (Note that in these H2+ or H2 examples, we are not "using" hybridization yet.)
When you consider covalent molecules formed by atoms with electrons in different types of orbitals, e.g. s and p, then again, it was found that a reasonable approximation for the solution of the molecular orbitals was obtained by a) combining the wavefunction of the different orbitals in each atom (the ones with similar energy, symmetry, etc.) to calculate a "hybrid orbital" and b) combining the hybrid orbitals of each atom to get an approximation of the wavefunction of the molecular orbits.