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.
But did it work because of the style or in spite of it? No reason to believe it wouldn’t be even more successful if he had been less abrasive like he is now.
Quantum mechanics didn’t supersede electromagnetism. Again, they’re different things. Electromagnetism is a fundamental interaction. Whereas quantum mechanics describes the mechanics of quantum particles. Whether those particles are affected by electromagnetic forces or not. It’s a description of how they behave at quantum scales.
Coulomb’s law has nothing to do with quantum mechanics, it’s a description of how macroscopic charged particles interact. What the OP should have said to be correct is:
Awesome to see the similarities between: Newton’s law of gravitation and Coulomb’s law