The Taranushi is a tradition of the Marid tribe once a djinn turns 14 years old. Each person goes to a place of their own choosing and grant three wishes to a deserving person. To prevent obstruction from outside sources, the initiation must be kept a secret. John and Philippa both undergo this ritual during the sixth book. John goes to Bumby with Groanin while Philippa decides to go to Pompeii in Italy.
Part of the reason it is considered a becoming-a-man (or in this case djinn) initiation is because it reflects their judgement in choosing a deserving person for the wishes and must consider the ramifications of granting wishes. They can't be thoughtlessly given to someone who might misuse them. As Nimrod wisely points out, the humans can be unpredictable and greedy. If the adjudicators (panel of judges) decide that the wishes are not justified, the young djinn can have their powers taken away for a year. This is all very serious business.
The tradition originated from the first great djinn known as Taranushi. When he was taken over by evil djinn, the event was to commemorate his overthrow by evil djinn. It is known to have evolved since its first origination, where at one time it would have involved walking across live coals.