The fundamental misconception is that we have natural genetic interests based on shared genes. Evolution does not create entities with genetic interests. It creates reproducing machines. Those are different things.

This is the fundamental point, for sure. But it is also true that I care more about a random human than a random banana. Evolution didn't completely fail to build instincts that approximate causing us to care about the prevalence of our genes in the world. I disagree with the assertion that it's so difficult to define genetic interest in a world where we share so many genes with so many species. It's only difficult if you try to define specific cases in isolation. What we actually do in life is make choices between alternatives, and it's pretty easy to define which of a set of alternatives is genetically closer to me than another. Naturally I don't have biological senses to measure and analyze genomes , so I couldn't use this metric in normal life even if I wanted to, but the idea isn't that much more complicated than trying to define which colors are more similar in a world where the visible spectrum is so much narrower than the larger electromagnetic spectrum.

Of course, I will also often care more about a random dog or cat than a random human. I can accept intellectually that this tendency is because our ancestors (mostly unconsciously) domesticated them to have exaggerated childlike features, and because I instinctively want to protect the innocent or helpless little ones. And under the right conditions I have no problem feeling sad for a random lamp. I think a lot of the problems people have understanding evolution come down to not realizing just how dumb a selection process can be and still yield incredibly complicated results, and to not being willing or able to reason about how contingent approximations break down out of distribution (feel free to draw the obvious parallels to other selection processes, even if most people around you won't).

From the linked post: 

Society is not biology. It is another level of order that we create on top of the psychological and biological.

This is true, but becomes less true the further back in time we look, when societies were smaller and less diverse. It used to be a very reasonable proxy. I also think it's worth acknowledging that historically, one of the common ways humans have tended to overcome tribal tendencies and form larger cooperative units has been by accepting myths of fictive kinship with rulers or deities. When we think of people as kin, even distant kin, it becomes easier to cooperate with them. When we share more language and culture it becomes easier to believe those myths. It becomes more reflexive, less dependent on considered choice and calculation. It is possible to break and redirect that tendency, for good and bad, but the default tendency is for these layers of organization to be linked.