There are many ways to do animal voice wrong. Most of these ways involve overthinking it, twisting your mind too hard around what animals would know: "The human jangled the small pieces of door-opening metal. Maybe I would get to take a ride in the big moving thing!"
There are many ways to do animal voice right. Most of these ways involve creating a set of rules for what these animals understand and owning those rules, whatever they are. Elephant and Piggie are almost indistinguishable from humans. Wilbur, Charlotte, and their friends have distinctive but mostly human-like voices. Despereaux tells Princess Pea that he honors her, and Princess Pea understands.
But the best animal voice I've read recently came from an unlikely source: a YA dystopia. In Patrick Ness's Chaos Walking trilogy (thanks for the rec, everyone!), thoughts are audible, so it follows naturally that we hear from animals as well as from humans. The animals are kept animalistic, and their self-expression is kept simple and thus believable. "Hungry, Todd." "Thirsty, Todd." "Boy colt?" Their words, and especially their identifications of the people closest to them, combine with their actions to tell us plenty about them. We learn enough to care about them as what they are - animals whose emotions are basic but include loyalty and love.
"Lap, Shosh," a certain cat seems to be saying. Signing off.