It does not sound like a fungal infection as there is no relation to the cold in this condition.
Raynauds is an abnormal reaction to the cold in extremities ie feet, hands, ear lobes etc and the 'primary' type of the condition can happen at any age. . The blood vessels spasmodically contract reducing the circulation to the area especially if cold areas have been touched or the room is cold, so keeping warm and maintaining the temperature around the person is important.The colours and tingling afterwards happens as the circulation is restored when warmer. Anxiety has also been known to play a part. Some people go throughout life never having any form of diagnosis and no form of treatment apart from keeping warmly dressed. I would think that a referral to a paediatrician would be helpful especially as there is pain involved and to put your mind at rest. Speak to your GP again. Hope this helps