As someone who plays Volibear and Shyvana a lot, I can tell you that Volibear does WAY less damage than either Shyvana and Udyr. Shyvana, for example, has enough damage to 3-shot squishies and enough %HP damage to shred the tanks. Volibear does just enough damage to barely kill one squishy IF he's not kited, then has to help peel/intimidate/tank for the rest of the fight. He makes up for it by having more CC and engage.
He's like Garen. He does just enough base damage to kill an unprotected carry, which is his job as a bruiser-tank. If he can't do the simple job of killing one target, then he's absolutely worthless. Think: Why would I play a bruiser-tank (like Volibear, Garen, Mundo, or Olaf) when I could instead play a high-CC tank (like Sejuani and Amumu)? Well, I'd only do it because the team needs a big frontliner who can scare the enemy carries away or kill them, instead of CCing for the backline. I might not get to do nearly as much damage as Zed or Yi (or Shyvana), but at least I get to be tanky and big while I do it.