Can Newton’s law of gravitation be derived from Coulomb’s law?

those two laws look similar because they both describe the propagation of a long-range field through three-dimensional space which produces a force that acts (in the simplest example) between pairs of objects, and in which the strength of the resulting force depends on some extensive property of each (charge in one case, mass in the other). the long-range-force part in three dimensions furnishes the 1/r^2 part, the influence of the extensive property furnishes the product m1 x m2 or q1 x q2, and the constant term (G or E) contains the fundamental strength of the coupling.

However, if we are talking about coulomb's law and newtonian gravity, those different fields do not couple or mix- which means you can't "make" gravity out of electrostatics or electrostatics out of gravity. Furthermore, gravity is always attractive because there are no negative masses, whereas electrostatic forces can be either attractive or repulsive because charges come in either + or - form.

This is a simplistic explanation. Note that there are far deeper reasons rooted in the underlying mathematics for why these things are the way they are and I invite the professionals here to weigh in on this.

To the best of our knowledge they are not deeply related although there is a theory called Kaluza–Klein theory that tried to interpret electro-magnetism as curvature of space-time much like gravity. There are, however, no real indications that this is correct.

To get back to the original question the relation is that the force equation has identical functional form with just different constants. This can be interpreted as coincident but is useful in mechanics since you can reuse many results for gravity in the case of charges that interact.

What if mass had a sign?

There are (let's keep it simple) Sun, Earth and Moon.

Earth goes around the Sun, so they have different signs.

What about the Moon? If it's attracted to the Earth, it would be repelled by Sun, and vice versa. This is not what happens.