Fuel injected engines have a "cold start" system. Basically, there's a temperature switch: ON when cold, OFF when warm, a relay or similar that is ON longer as the temperature falls and finally an electric valve that sprays a bit of fuel into the cold engine and so enriches the air / fuel ratio for cold starts. The valve is ON only for a matter of seconds -- just to get the motor going.
One of these parts is either disconnected or has quit.
That's what's wrong with your car.