The similarity of artificial neural networks and the human visual cortex goes very deep, and in many ways the human visual cortex was the inspiration for the techniques we use for the design and implementation of ANNs designed for image recognition. So in that direction, the similarity seems obvious to me.
The reverse direction, though, is a question about how the human mind works under the influence of LSD, which you'll probably get a better answer asking about in the biology or cognitive science stack exchange sites.
Some brief details to add to the answer, though: the human visual cortex is arranged in layers that correspond to increasing layers of abstraction. In the eyes themselves, photons are detected by light-sensitive cells and added together to make what are essentially the color elements of pixels. Those are then routed to another layer which does something like edge detection, and then the next layer does something like shape detection, and so on up to higher level concepts like "a cat's face.' If LSD lowers the activation threshold for those neurons, or makes them more excitable, then more things will be interpreted as having the higher level concept (and so a patch of rough texture may have a face jump out of it, for example).
The way that CNN "deep dreaming' works is that the base image is amplified. That is, to make a particular patch look more like a dog, the shapes are nudged to be more dog-like, and the shapes nudge the edges, and the edges nudge the pixels.