Mine is more a technical curiosity, because the $\text{Laplace operator}$ is one of the many differential operators from which we can start, we also have the $\text{Jacobi Theta}$ $\theta$ operator $\rightarrow$ also that is an elliptical function (also Ramanuajan generalization preserves its properties).
Why start from Laplace (homogeneous Poisson equation)?
What's so special and more than other differential operators for example compared to that of Jacobi?
For example, if I started from the $\text{Whittaker equation}$ or from the $\text{Bessel equations}$ (ordinary linear equations of the second order homogeneous) what changes for a physicist that deals with potentials and harmonic functions?
The equations that physicists use encode physical 'stuff' which is required to respect the symmetries of the universe. In simple models, we assume space is just $\mathbb R^n$ and the symmetrices are the Euclidean motions.
It turns out that one can describe explicitly all differential operators that are invariant under the Euclidean group. They have to be translation invariant, which is easy, and then invariant under the orthogonal group, which is a bit more complicated. It turns out that the Laplacian is the simplest invariant differential operator — the only one of order $2$.