There is another issue with car batteries (and any other chemical batteries). You can't just dump all that energy at once, even if you short circuit them. The chemical reaction needs time to happen. You would actually need capacitors for that. And the problem with capacitors is, that they can't store a lot of energy.
Perhaps the 'clip' is actually a smaller chemical battery, custom made by X-Com, and a massive bank of capacitors.
The battery could store more energy than you need to do two shots, but the two shots are all that fits into the capacitors, and they would need too long to recharge. The battery would then be there just to keep them topped off.
No, you don't need large capacitors.
From the wiki of Ignition coil
https://en.wikipedia.org/wiki/Ignition_coil An ignition coil (also called a spark coil) is an induction coil in an automobile's ignition system that transforms the battery's low voltage to the thousands of volts needed to create an electric spark in the spark plugs to ignite the fuel.
So a taser powered by car battery and ignition coil can do dozens of high voltage hits. Hitting a normal human with the full power will likely kill him on the spot, but we can imagine that the agent controls the output with some knob.
As for the delivery method, I assume it is the same for the normal taser: a bunch of wires with arrow heads are launched at the target and make contact. An arc lightning will hit the closest uninsulated surface - usually the agent holding this contraption unless he wears protective gear. In the later case it will arc to the ground.
So yeah if you want to be realistic the Taser Cannon is way too underpowered (either in power or in number of hits per battery) for what we could do with the 90's technology.
I am not complaining actually, hitting that Black Lotus Avatar with it was very satisfying.