I think that it's not so much that fighting types are too common so much that they are so prominent. 5 total types are weak to fighting, so chances are, on the average playthrough, players will have one or two Pokemon weak to it, maybe more. On top of that, they often have nasty attack stats, so when you take a hit from one, it will hurt if your Pokemon doesn't resist it.
On top of that, it's a popular type to stick onto an evolution to make it cooler:
"How can we make this tadpole cooler and stronger? Let's make it a Fighting tadpole!"
"How can we make this fire chicken more awesome? Make a kung fu fire chicken!"
"How can we really make this attack-based Mega Mewtwo better? Half-Fighting, of course!"