The relation between them is that they're both forces that scale with the inverse square of the distance between the objects. Any force that scales with the inverse square of distance has pretty much the same general form.

Another similarity is that both are incomplete, first approximations that describe their respective forces. The more complete versions are Maxwell's laws for electromagnetism and General Relativity for gravity.

There's some relation in that they both act on fields, but the things that affect those fields are very different (higgs bosons and electrons respectively) and the relationship between all that for an 'unified theory' is a topic of much research. IANAP though (not a physicist)

The most accepted theory among physicists is that "shit's crazy, yo".

Electromagnetism and gravity are both mediated by massless bosons; photons and gravitons respectively. This is why both forces follow the inverse square law.

There is one thing particularly interesting, and that is that the inverse square laws appears again. It appears in the electrical laws for instance.

That is electricity also exerts forces inverse to the square of distance with charges. One thinks perhaps inverse square distance has some deep significance, maybe gravity and electricity are different aspects of the same thing

...

Today our theory of physics, laws of physics are a multitude of different parts and pieces that don't fit together very well. We don't understand the one in terms of the other. We don't have one structure that it's all deduced we have several pieces that don't quite fit yet.

And that's the reason in these lectures instead of telling you what the law of physics is I talk about the things that's common in the various laws because we don't understand the connection between them.

But what's very strange is that there is certain things that's the same in both

Richard Feynman and 45:48 https://youtu.be/-kFOXP026eE?si=hAIvDhWVGxMOvEi1

It's really simple, they are both radial fields with a 1/r potential, thus a 1/r² force. Newtonian gravity is just a weak field approximation of general relativity, where you have very different equations, for example Einsteins field equations.. One electric charge creates an electric field, and another charge will interact with it, but the motion itself still depends on the mass of the second charge. Matter instead curves spacetime itself, and the curved spacetime tells matter how to move. Source: MS in physics.

Doubtful but interesting thinking. It’s actually a rather simple equation that explains how two equally weighted forces affect one another over distance. The numerator expresses that both forces carry equal weight in the interaction (if they are both the same kind of force, eg gravity or electromagnetism, this makes sense) and they are constructive interactions (both add to the intensity of the interaction) hence multiplying one by the other. The denominator just indicates that the distance between the two things exponentially degrades the force at a power of 2, since the force is spreading out in 2 dimensions (imagine a cone starting at one point and extending to the second, so that when you reach the second point the force is spread across the cross section of that cone, but the only part of the force affecting that second point is the part that touches it).