Is there data to show that pets actually cause more damage? I see this argument get thrown around quite frequently (often by landlords), but I don't buy that it's a major issue.
That's not to say that pets can cause tons of damage: if someone let's a cat spray inside and doesn't immediately clean it, that can cost tens of thousands if it seeps down to the underlayment. But (1) that ain't getting covered by the $50 - $100 they want to gouge people for; and (2) tenants are liable for that either way.
It isn't that every pet causes more damage as a rule, its that very frequently pets do damage that is quite expensive. Urine can require floor and subfloor replacement. I've seen dogs chew through walls, etc. Pets in common areas leaving tracks and fur, etc. I'm a dog lover, but it would be folly to pretend that owning a dog does not cause additional wear and tear on my house.
That said, damage from pets should be handled through the damage deposit. Pet rent is just another way to pad the bill.