What is the current state of the art in animal facial recognition? As far as I can tell, most work on facial recognition has been done in relation to human faces. Has any significant work been done for dogs, and are there any special challenges that would make it difficult to adapt existing work to this use case?

To clarify: I don't mean image classification, but segmentation. In other words, I refer to a model that can take an image that can have any number, including zero, of dogs in it, and that will produce bounding boxes around each dog's face.
In case of animal classification - trying to specify the animal, trying to tell cat from a dog - it should be similar to human faces. An algorithm such as SVM could be used in that case. Trying to tell two breeds of dog apart would be more challenging. I have come across this article explaining how they use facial recognition software for animals news. And I have also come across  this article where they write about a novel method for identifying individual cetaceans using facial features and symmetry on dolphins.