There is no "center of charge" that simplifies calculations completely because there is no center gravity that does this, either.
Two rigid bodies feel gravity that is quite similar to that of point masses, but not exactly the same. Unless the objects are perfect spheres, they feel tidal forces, which depend in second-order and higher derivatives of the potential.
We can only calculate gravitational interactions as points at the center of mass if we want to ignore these second-order effects. On a day-to-day basis, using gravity for things like playing baseball with your nephew, that's fine. The tidal forces are usually very small in applications that we encounter because the phenomena are small compared to the characteristic length scales involved (e.g. a baseball's trajectory is small compared to the radius of Earth, Earth's diameter is fairly small compared to the Earth-Sun distance, etc.)
We can indeed make the same first-order approximation in electrostatics that we do in Newtonian gravity. In practice, a slightly different procedure is normally used, called the multipole expansion. First, a reference point is fixed. The total charge, called the monopole moment, is independent of this reference point, so if you want your first-order approximation to be good, it is wise to choose a reference point near the center of your charge distribution. Higher-order effects are then calculated relative to the reference point by computing dipole, quadrupole, and higher moments of the charge distribution. These higher orders are more likely to come into play in everyday life in electrostatics than in gravity because the size of the charge distributions is similar to the separation between charged bodies (e.g. two balloons you rubbed on your head and are playing with are not very far separated from each other, compared to their diameters).
You might also be interested in the hard sci-fi novel "Incandescence", in which a species of beings living inside a large asteroid orbiting a black hole observe tidal effects to discover general relativity without ever seeing the outside world. It's an interesting demonstrations of what higher-order gravity effects look like beyond the center-of-mass approximation.
The author, Greg Egan, has a web page explaining the tidal effects described in his novel here.