Why does Zilean hate armored bears (Volibear)?
Solution 1:
This is a developer in-joke, similar to how the 3 ninja champions obey the Inverse Ninja Law, or that Leona's passive deals 1 less damage to champions with sunglasses.
In this case, you first have to know that Zilean is named after developer Tom "Zileas" Cadwell, and Volibear is named after Travis "Volibear" George. These buffs are tongue-in-cheek references to a long-standing forum faux-feud (or maybe real feud? Who knows?) about the plausibility and desirability to add an armored bear champion to League of Legends.
Zileas was against it, and Volibear was (clearly) for it.
It's pretty crazy.
The Forums got pretty riled up too.