How much resources are really saved by this process?
Very little. In fact, so little that for example Toyota that used to have a stop/start system in they Yaris small car got rid of the system (probably because it requires beefied up battery and starter motor that are more expensive than in non-stop/start systems).
My journey to work is driving at about 50 km/h average speed but 20 km journey takes half hours. So I assume a commute has about 6 minutes of idling.
A car uses 0.7 liters per hour idling. So 0.07 liters of fuel is saved for my commute. At fuel consumption of 6.5 liters / 100 km, the commute takes 1.3 liters of fuel. That's only 5% savings. In contrast, we have technologies like hybrid vehicles that allow far greater reductions in fuel consumption.
Does the car need to be designed differently to make this effective? For example, do they make the ignition system different?
There are several differences in normal cars and stop/start cars.
Firstly, a stop/start car has a beefied up battery. In higher end vehicles it may be expensive absorbed glass mat battery, but cheaper stop/start vehicles use enhanced flooded batteries that have plates withstanding more use at slightly lower states of charge than 100% full and withstand cycling better. Also the stop/start system has a beefier starter motor that is rated for more cycles than on non-stop/start cars. Also usually the system has a counter that counts how many times the stop/start motor has been operated and tells the driver when to replace the stop/start motor to resume the stop/start functionality (when the lifetime is over, it starts to work like an ordinary non-stop/start car).
Furthermore, usually stop/start systems have engines with variable valve timing in intake and exhaust sides that allow minimal effective compression ratio at the time the engine is restarted. This makes for a very smooth restart process by minimizing the amount of energy needed to restart the engine.
The HVAC system also has improvements. For example, in stop/start cars the blower motor is often automatically controlled to have minimum possible speed during engine stop (this may be omitted in manual HVAC systems), and also there is automatic engine restart if the the temperature inside the car becomes too hot (requiring AC which requires engine to be on) or cold (requiring heat from engine).
Furthermore, the battery is saved by a low voltage detection circuit that automatically turns the engine on if the battery voltage becomes too low.
Can we do the same manually (just remember to shut down when stopping, and start again when the light turns green) and get similar benefits?
No. You'll wear your starter motor and your battery if doing that. Normal batteries aren't designed to continuously cyclically produce power for the lighting system when the engine is off, and normal starter motors will be worn out in no time if doing that. Also you lack the conveniences like auto-restart on too hot or cold temperature, and the essential feature which restarts the engine automatically if battery voltage becomes too low.
Also it's less convenient to not have the engine automatically restart when pressing the clutch or releasing the brake. Furthermore restart will be more rough because the car may not have dual variable valve timing minimizing compression ratio of restart.
The battery could obviously be replaced with a battery that withstands abuse, the type of battery used in stop/start cars, but you probably can't replace the starter motor with a stop/start variant because you may not find a compatible stop/start starter motor for your car that originally had non-stop/start starter motor.