I'm just going by what is said in the wiki and what happens in game. The wiki states: "Breed - 1 - passive - you have a [0-6] percent chance, depending on your level, of double drops while breeding animals". This would imply (since it's worded just like the mining, woodcutting, fishing, and herbalism skills) that ALL players can breed animals, but farmers would have a chance of "double drops" when breeding. What double drops means, not sure. Could mean 2 babies from 1 breeding, or double exp from breeding, I don't know. But if it was meant to be that only farmers could breed animals, then the skill would simply say "Breed - 1 - passive - you can breed animals".
If we're going by the wiki's description of the skill, WHO can use breeding is working as it should, the problem is that farmers apparently aren't getting the bonus to breeding that the skill is supposed to give.