Question

In what ways are electromagnetic force and gravitational force similar

Answer 1

In particular the gravity is a theory very different from any other field theory. The electromagnetic force, the nuclear weak force and the nuclear strong force are all described in terms of an exchange of particles. It means that the distant action is in reality the transport of a particle from the source to the destination.
So for example two electrons that repel each other, are exchanging photons to create the force.

Answer 2

From looking at the two force equations, you can see the similarities and how gravitational force can be considered parallel to the force between two charges.

Besides being proportional to the inverse of the square of the separation: both forces extend to infinity,They also both travel at the speed of light.