Sure, we can apply even higher level features, and we can generalize from a picture of a cat to a black&white or text description of it.
But we still memorize a lot.
That last bit about "multiple levels" is key. Shallow models like kNN, SVMs, Gaussian Processes, etc. don't do that; they learn to recognize/memorize patterns at only one level.
Second, it's not PR! This stacking of layers, when done right, can overcome the "curse of dimensionality." Shallow models like kNN, SVM, GP, etc. cannot overcome it; they perform poorly as the number of input features increases. For example, k-nearest-neighbors will not work with images that have millions of pixels each.
Third, I'm only scratching the surface here. There's a LOT more to deep learning than just stacking shallow models.
Perhaps deep networks work well because they learn to memorize "the most common patterns of auto-correlation" they see in the training data at different levels of function composition.
In fact, we do this explicitly in convolutional layers, which by design learn to represent every input sample as a combination of a finite number of fixed-size square filters.
...and the reason why deep networks might be "generalizing" so well is because not all distributions of natural data are equally likely!
In practice, objects with the same or similar labels tend to lie on or close to lower-dimensional manifolds embedded in data space.
...and this concentration of natural data distributions might be a result of the laws of Physics of the universe in which we happen to live: https://arxiv.org/abs/1608.08225
So, yes, deep learning could very well be a really fancy form of memorization.
I hadn't thought of it in this way before. Very interesting :-)