However, the charging systems and voltage regulators of the two are very different, and they deal with over/under voltage in a way that is not good for bike charging systems.
More accurately, you'll probably be fine if the car is running, but you gain nothing by doing so because the car's battery is SO much bigger than the bike's, and there's potential for damage to the bike's charging system, so it's not worth it.
A car's system regulates voltage by varying the input to the field coil in the alternator. If the regulator wants more voltage, it pumps more electrticity into the field coil, and more voltage comes out.
A bike has fixed magnets for the field (instead of windings like the car) so the input field can't be changed. Excess output voltage (power really) gets dumped into the air by way of the voltage regulator which simply turns the extra power into heat. This is simple and lightweight and perfectly adequate for a bike's electrical system.
However, when you hook up a running car to a bike the two charging systems fight (because statisticly there's no way that they can both be aiming for exactly the same voltage). When they fight, the bike looses, big. If the car thinks the voltage should be 0.1 volts higher than the bike, it will keep pumping more power into the field coil to try to raise the voltalge. The bike, on the other hand, will try to get rid of that 0.1 volts by turning it into heat through the voltage regulator.
The car has a relatively huge alternator and engine, so its ability to dump power into the joined system is greater than the bike's voltage regulator's ability to disipate heat, and it burns up.