If a pet owner is walking a dog on a leash but isn't watching the dog's every step because she's also holding her child's hand, and someone runs a red light and hits and injures the dog, causing thousands in medical bills, that's a very different insurance situation from a pigeon.
The pigeon is a nonentity in the accident. It's as much at fault, or not, as a tree. And it doesn't need medical or cremation bills paid. Cleanup is handled about the same way as debris from a car.
In a situation where the dog or owner isn't negligent but the driver is, it's useful to talk about whether the dog was on the leash, in the crosswalk, has right of way, is walking with the walk signal on, and is visible. You might prefer to leave the dog out of it and talk about the owner, but I could see myself wanting to get straight to the point and talk about the dog. I imagine lawyers also might be inclined to cut to the chase and talk about whether the dog was following the rules of the road.