A capacitor should discharge more quickly than a battery. What happens is some of the charge is stored in a capacitor so the battery has to work less. I google translated the japanese webpage and they mention quicker starting time. But judging from the size of this box, the capacitor is probably not that big. When you think about how much current is discharged by a battery, the capacitor will probably have no effect. They are attempting to minimize the inrush current coming out of the battery from what I can see. Do they have a size on the capacitor?

They also bring all the grounds to one point on the negative battery terminal. This is considered good grounding practice to minimize ground loops. The way autos are grounded now, the grounds could have different voltage levels.