If you take a pre-trained convnet (which, by analogy is like a person who has had 'life experience' of looking at objects), and extract activations for unseen object categories, in many cases you CAN one-shot-learn these new object categories. Try feeding them into a SVM or use L2 distance between test images and the one-shot exemplar image.
On top of this, there's a lot of work on memory-augmented nets and meta-learning for learning new categories on the fly.
For example, with bears -- I personally know of black bears and polar bears. I can be a little more detailed with fish but with dogs there are dozens of "different" [easily recognizable] types within the same category of "dog".
Of course, the disparity between deep neural nets and human brains remains unknown. A human learns the difference between a cat and a dog, while at the same time, learns so many different things, yet a neural net only learns the difference between a cat and a dog. We don't know how much we don't know.
I think the SOTA is probably [2], which came out of DeepMind. There's still a way to go before it matches ResNet performance on ImageNet (or even human performance on any real task) though.
I wonder if they can also address the following problem. Currently, deep learning toolkits need thousands of training images to classify images of, e.g., dogs and cats. A human, in contrast, could learn the difference between a dog and a cat by looking just at a single example (or perhaps a few). So right now, deep learning is too much "simple" pattern matching, and too little real "AI".