Animal logos work great for many reasons.
1. We tend to associate certain human characteristics with animals. These associations can work well to build a certain brand image. For example the fox is smart. A bear is strong and wise. A bee works hard. A dog is loyal and friendly. Etc. No wonder great writers like Orwell and Andersen used animals to dramatize human traits in their stories.
2. Animal shapes and faces are easy for humans to recognize and remember because we're hard wired to watch out for them in nature. Eyes, teeth and claws can be especially attention grabbing, but they can subliminally scare the audience away too, so these designs must be balanced by soft shapes and colors.
3. Cute characters evoke maternal/paternal feelings. We can't help but want to cuddle and protect a cute creature. We have instant connection to baby animals. Such strong emotions are extremely hard to evoke with non-animal designs.
4. Animal symbols are easy to extend to other media, like soft toys, keychains, balloons, etc. They are also a good starting point for a campaign that can be built around such a mascot.
Here are some examples of great animal logos: