Not imaginary creatures like mandrakes (Sophie Quire and the Last Storyguard) or jinn (The Eye of Midnight) or chamelons (The Secrets of Solace), but rather animals that talk or communicate with humans or take on anthropomorphic characteristics:

Bats
Forest of Wonders by Linda Sue Park.
Shadow Magic by Joshua Khan.

Foxes
Maybe a Fox by Kathi Appelt.
Furthermore by Tahereh Mafi.
The Night Parade by Kathryn Tanquary.
Pax by Sara Pennypacker.

Hamsters
Hamster Princess: Of Mice and Magic by Ursula Vernon.
Time Traveling with a Hamster by Ross Welford.

Squirrels
The Tale of a No-Name Squirrel by Radhika Dhariwal.
The Magic Mirror: Concerning a Lonely Princess, a Foundling Girl, a Scheming King and a Pickpocket Squirrel by Susan Hill Long.
Evolution Revolution: Simple Machines by Charlotte Bennardo.

Rats and Mice
Armstrong: The Adventurous Journey of a Mouse to the Moon by Torben Kuhlman.
Word of Mouse by James Patterson and Chris Grabenstein.
Brightwood by Tania Unsworth.
The Rat Prince by Bridget Hodder.
A Tail of Camelot (Mice of the Round Table #1) by Julie Leung.

Dogs:
The Poet’s Dog by Patricia MacLachlan.
Foxheart by Claire LeGrand.
Making Mistakes on Purpose (Ms. Rapscott’s Girls) by Elise Primavera.
The Inquisitor’s Tale: Or, The Three Magical Children and Their Holy Dog by Adam Gidwitz.
Behind the Canvas by Alexander Vance.
The Wizard’s Dog by Eric Gale.

Tigers:
Sophie Quire and the Last Storyguard by Jonathan Auxier.

Cats
Fortune Falls by Jenny Goebel.
The Nine Lives of Jacob Tibbs by Colin Busby.

Wolves
Evil Wizard Smallbone by Delia Sherman.
This Is Not a Werewolf Story by Sandra Evans.
The Peculiar Night of the Blue Heart by Lauren DeStefano.
Simon Thorn and the Wolf’s Den by Aimee Carter.
The Wolf’s Boy by Susan Beckhorn.

Rabbits
Ember Falls by S.D. Smith.

Sharks
Stingray City by Ellen Prager.

Pigs
Liberty by Darcy Pattison.

Bears
The Growly Books: Haven by Philip Ulrich.

Hyenas
The Bolds by Julian Clary.

Skunks
The Midnight War of Mateo Martinez by Robin Yardi.

Snakes and Other Reptiles
Outlaws of Time: The Legend of Sam Miracle by N.D. Wilson.
Dragonbreath: the Frozen Menace by Ursula Vernon.

Dogs win, with wolves, and rats and mice coming in a tied second. If you or your child have your own animal avatar or interest, you just might be able to pick a recent book related to the animal of your choice.