Not just heat dissipation, but, consider why it's not just a round bar?
You are trying to very accurately measure the infinitesimal voltage drop across a huge conductor, with an infinitesimal known, precise, calibrated amount of resistance, blindly.
That is, you build the shunt to a spec, or you measure it afterwards, and expect the resistance to be that precise amount. The calculated voltage drop across that very precise resistance standard is used to extrapolate and tell you what it thinks the current should be according to those parameters. But it's blind. It never measures the resistance, it's calibrated for what it's expected to be.
Resistance varies hugely with temperature. For extreme example, when a lightbulb starts up, cold, it has 1/16th the resistance and therefor draws 16x as much current as when it's hot (and bright).
If you calibrated a lightbulb as your shunt, and measured it when cold, you would be reading, for example.... 16 amps when there's only 1 amp flowing (a 120W bulb).
In the case of the car, your gauge might show you're drawing 1600 amps when you're only drawing 100.
Now obviously the shunt isn't getting glowing-white-hot, that's an extreme example, but a few degrees difference on an already-so-small-it's-hard-to-measure voltage drop will make your gauge lie to you. Maybe not 1600%, but, how unsatisfied would you be with even 10% or 25% inaccuracy?
That's why shunts are always thin, wide bands, or multiples thereof. Lots of surface area relative to their cross section.
It's not because you're worried about heat build up or inefficiency, it's because it's a highly calibrated measurement device that needs to be as consistent as possible.