As car batteries provide only 12 V instead of ~230-250 V, I assume it takes much longer to charge them compared to a common wall socket?
The critical element between the power supply and the phone is the charger. Most phone chargers are at 5V and even the USB 3 they provide (at the time of writing) current of about 3 Amps. This means a maximum power of 15 W (=5V * 3 A)
The car batteries maybe at a lower voltage (V) but they can draw more current (I). (Even though most car batteries are only 6 or 12 volts, a 12-volt battery can produce are said to produce several hundred amps -- according to solar Mike even 1000A). Thus the power they deliver is $V\cdot I$, and its significantly higher than 15W.
Of course, the battery power output cannot be compared to the grid, however there is still more than enough capacity for the charger (at least theoretically).
- The "charger" or charge controller is in the phone.
- The external device is a power supply.
- On connection by USB the phone will negotiate with the charger to see how much power the charger can supply.
If your car's power supply doesn't support negotiation you may be limited to a charge current of 0.5 to 1 A and charging will take more time.
Newer cars have USB adapters built-in. This adapter can charge faster and can deliver higher wattage to small laptops too. They can also transfer data such as music and video from the phone to the screen on the dashpot.
The old adapters come in different ratings, 1A and up.
A good adaptor can charge a cell phone almost as fast as the wall outlet.
Some of the adapters have cheap connections that need cleaning and wiggling in to establish a solid contact. Some SUV cars have extra cigarette sockets with higher ratings for camping lights and small electric tools.