For what it's worth, my instinct tells me that this is primarily an aspect of temperament you're born with. Either you have a strong emotional reaction to the idea of eventual nonexistence or you don't. If you do, religion is the most common way of dealing with it. The particular religion you choose is obviously mostly cultural, passed down from your parents and peers the same way you learn a particular language as a child, but the motivating fear itself probably isn't.
But either way, does this really reveal something essential about what it means to be human? In one sense, yes: a knowledge that someday we'll die is unique to humans (though fear of death plainly isn't), and our response to that knowledge has been a defining feature of human cultures for millennia. Still, there are hundreds of other things that are unique to humans too, and I don't think there's any special reason to give this one pride of place.
I take the former point entirely. I find Kevin's final statement unpersuasive. To be human is to be aware of our own finitude, and to wonder at that. Montaigne argued that to philosophize is to learn how to die. Camus put it differently: men die and they are not happy. For me, this last thing is our first thing as humans. It is our defining characteristic, even though some animals may experience this in a different way.
And our ability to think about this casts us between angels and beasts. It is our reality. Facing it is our life's task.