Symbolism, Imagery, Allegory
There is so much animal imagery in this book that sometimes it's hard not to picture the characters as walking, talking animals, in the vein of Winnie the Pooh. Heck, the Todd family even lives at Fox Corner, which sounds an awful lot like Pooh Corner.
It Bears Repeating
Both Ursula and Teddy are named after types of bears. Ursula means "little she-bear" (4.49) and Teddy (Edward) is named after a strangely cuddly U.S president. We're not sure what being a bear has to do with their personalities, but Ursula and Teddy are very fond of each other and their names are a shout-out to their strong connection.
Keep it Foxy
The family's last name, Todd, means fox (so Ursula is kind of named Bear Fox). Foxes are seen as both good and bad. In one timeline, actual foxes eat the girls' pet rabbits, but in another, Maurice is vilified for shooting a fox on the property.
So why is the family named Fox, from a literary perspective? Well, it could be that foxes are seen as cunning and wily, and those two terms fit Ursula to a T. At one point, Ursula describes herself as "a fox without a hole" (26.194) when she completely loses her way in life. But in later lives, she's able to think her way out of that situation.
An Act of Dog
It's not all just humans with animal names. Life After Life is also populated with dogs given human names, like Trixie. Like family pets do, these dogs die. And although they're a part of the family, and their deaths are sad, they're always replaced by another dog, each one slightly different from the last.
This is a lot like Ursula's many lives. Sure, she dies and it's sad, but she's replaced by another Ursula, each one a slightly different copy. She shares other traits with the dogs, too, like Bosun, a French mastiff, "the kind [of dog] that pulled people from burning buildings and rescued them from drowning" (6.21). Ursula saves people from burning buildings in one life, and she herself is rescued from drowning.
When the family gets kittens, Pamela acts with "uncharacteristic melodrama" (26.59). This is the only timeline where she isn't the stable sister, the voice of reason. Insofar as the dogs represent Ursula, then perhaps the cats are a symbol for how essential Ursula is to the Todd family—when her symbol is out of the picture, things simply aren't right. Either that, or Atkinson is just more of a dog person than a cat person.