You know, it's right when people say dogs can't feel love or guilt like a human can. There's maybe 5 animals on the planet that can process those emotions, and those are pretty much limited to a limited set of primates and cetaceans.
That said though: Dog's aren't soulless things. They're not just following a routine. Dogs make very real familial bonds with their owners and families, they feel real loyalty and yes, it's an instinctual thing. That doesn't make it any less of a real connection with their human family. A need for loyalty and support from a group of trustworthy family members is something dogs and humans share on a deep level.
That dog isn't sad in a human context that his loved one died. But he is very much missing and pining for his lost family member in the deepest way he can. It's still something beautiful and sad.