So, what it unique in human social organization and social solidarity is humans' ability to relate "socially" to a wider group of their species than other primates like chimps and monkeys. It's not that other primates are not social, it is that humans extend that sociality even further. This wider group is across both time and space. I emphasize the social "solidarity" across time. Note that the ancestors reach forward in time, as well as the descendents reaching back in time. The invention of culture occurred when some hominids thought "lets communicate with future generations. How can we pass messages on to our future descendents".
[WS:] I do not think it is unique, but rather a matter of degree. Everyone who has companion animals (as opposed to pets) can attest that animals can to some degree empathize with other species. However, we know very little about their emotional states, because we have very limited ways of communicating with them. To cover up that ignorance, we uncritically accept opinions of autistic psychopaths like Descartes* who ex cathedra pronounce that animals do not have emotions well, because "I say so."
*) Descartes is believed to nail his wife's dog paws to the table and flay him while telling the audience to ignore the dog's cries as mere "automatic reactions" of the body (http://www.avam.org/exhibitions/home.html .) Talking about monkey-see-monkey do, this reminds me of a copy cat re-enactment of nailing a man to a stick, which became the founding myth of x-tianity. That makes it a glorious contribution of x-tianity to science, indeed.
Wojtek