Can you make Winston's jump work like Hanzo's?

No this sort of customization is not available for Winston's Jump Pack. It's unclear why you'd want it to be, aside from "put jumps on the jump button", so maybe you just need to think about it differently. After all, Hanzo's Lunge is the exception rather than the rule; pretty much all other characters' movement abilities are placed on the ability 1 button (default Left Shift).