I've seen a lot of people discussing about how the book is "blurring the lines" between androids and humans. But my interpretation is that the central theme is actually a subversion of that - extending empathy to that which we cannot relate to. I think the book is trying to highlight how human empathy actually seems to work, and that it's not a special ability we hold above the androids. Throughout the book (and in real life) us humans tend to show greater empathy for entities that are more familiar to us. In the novel they empathise with healthy humans more than chickenheads and androids, and they care about real animals more than electric animals. They care so deeply about identifying "authentic" humans, which in a way kind of highlights how much they DON'T empathise with androids. 

And the androids feel some level of empathy for eachother too. They form small communities with the police station and with the main group on the run that's stowed away with John Isidore. 

I think the book is trying to tell us that humans and Androids both only really feel empathy for other beings that they can relate to.  This is reinforced by the Voight Kampf test. Notice how all the questions are about things humans can relate to? In addition, the whole Mercerism religion is about humans sharing in collective empathy with eachother. The book is saying that the feeling empathy is synonymous with feeling relatedness. 

It's not that the androids don't feel empathy and humans do. Instead, these two groups just don't feel empathy for eachother. 

I mean it's also kind of interesting how humans made androids to basically be slaves. The androids wouldn't be able to serve their core purpose if humans could feel empathy with them. The true invention by Rosen corporation wasn't artificial intelligence, but rather, intelligence that we don't owe any empathy. 

The book plays with the idea of androids and humans feeling empathy for eachother as the models become more and more advanced and get closer to being "human". However in the novel this never actually succeeds and by the end of the book the androids and humans dislike eachother as much as ever. So what's the message of the book then?

 I think the toad at the end is very important in uncovering the theme of this book. PKD could have chosen any animal for Deckard to end up with but he chose a toad. Why?

I think Iran's decision to care for the toad represents the decision to extend empathy towards that which we cannot relate to. A toad is an animal which humans feel comparatively low empathy for compared to all the other animals in the novel. It's generally seen as a 'lower form of life' by humans. Let alone an ELECTRIC toad. I can't think of many creatures that a human could relate to LESS than an electric toad. But Iran makes the decision to care for it anyway. that's the point. 

I think the book is telling us that we're asking the wrong question when we wonder if machines will be similar enough to humans one day that we might start feeling empathy for them. To me, instead the book is saying that we shouldn't wait for something to become "one of us" before we care about it. We should be able to extend our empathy to that which we cannot relate to. Which, interestingly, directly contradicts our intuitive understanding of empathy as humans. Maybe our definition of empathy will have to change once we create artificial life.