There's a scene around the middle of the book where some of the workers coerce Candy into letting them shoot his dog, who is old and feeble. They do so, and Candy expresses his regret later when he says that he should have shot the dog himself. You shoot your own dog, he says. It is a foreshadowing moment.
In a sense, Lennie is George's dog. At the end, when Lennie must and will be shot by one means or another, George knows that he must be the one to do it, for not only is Lennie George's responsibility, but George is able to do it in the manner least painful for Lennie. And indeed, George does just that. So as we can see, it is not at all inaccurate to say that George shoots "his own dog" at the end of the book. That's precisely what he does, even if that's not the most literal way to say it.