Question

Why do we multiply wave functions in a multi electron system?

Answer 1

The particle exists, the wavefunction is a solution of the quantum mechanical equation specifically defined for the boundary values of the observation. Its complex conjugate square gives the probability density of finding a particle in a specific location.

For two wavefunctions to meet, it means a different set up of the problem, possibly new potentials and boundary conditions will define a new probability distribution.

Many body systems are not analytically solvable in classical physics. The answer by CuriousOne describes the mathematics developed to describe the quantum mechanical sol