What makes us human is the same as what makes koalas koalas, orcas orcas, or cedars cedars. It is is the particular set of inherited traits that set the parameters for our engagement with the ecosystems in which we are embedded. Feeding is surely one of the key modes of interaction between organisms in ecosystems, so in light of our evolutionary heritage as generalist omnivores, hunting is surely one of the key aspects of the ecological vocation of human beings.
I suppose the question about hunting making us human was asked with aspects other than the ecological in mind. Many humans in Western culture now see themselves primarily as economic agents, and perhaps as intellectual or social agents. Only a minority (mainly women) really see themselves as biological entities, and even fewer truly see themselves as ecological agents.
The emphasis of the economic and intellectual over the biological and ecological, and the narrative that this represents “progress”, would tend to lead to the idea that hunting is not or is no longer an important human characteristic. But it’s this very schism between mind and body, between economy and ecology, which has led the West to perpetrate the current ecological crisis.
That’s a value system that needs to be abandoned, if we stand any hope of halting or reversing the accelerating disaster. We will need to value nonhuman nature and also embrace our human biological and ecological embodiment. Among other things, that will mean direct, mindful participation in ecosystems in the modes evolution has equipped us for -- including hunting.