Three is the smallest number of points that fixes a plane and the smallest number of straight lines that can enclose a figure, and those two facts do more symbolic work than any doctrine about the number. A structure of three has a beginning, a middle and an end. Two things are merely opposed; three things make a pattern. That is why triads appear in traditions with no contact between them, and why the recurrence is weak evidence of anything shared.
The triads are genuinely everywhere. Egyptian religion organised its major cults in threes, Osiris, Isis and Horus at Abydos, Amun, Mut and Khonsu at Thebes, as a deliberate theological device rather than an accident of family relations. Hindu tradition gives the Trimurti of Brahma, Vishnu and Shiva, a scheme that is comparatively late and never displaced the sectarian preference for one of the three. Greek religion gives the three Fates, the three Furies, the three Graces, and a cosmos divided by lot between Zeus, Poseidon and Hades. Norse tradition gives three roots of Yggdrasil and three Norns.
The Christian Trinity is a different kind of claim and should not be filed with the rest. It is not a group of three deities with separate portfolios but the insistence that Father, Son and Spirit are one God, worked out against strenuous opposition over four centuries and stated at Nicaea in 325 and Constantinople in 381. Trinitarian theologians have generally resisted comparison with threefold patterns elsewhere, precisely because the point of the doctrine is the unity rather than the number.
The threefold repetition of folk narrative, three brothers, three wishes, three attempts, the third succeeding, is a structural feature of oral storytelling, identified by the Danish folklorist Axel Olrik in 1908 as one of his epic laws. It belongs to how tales are remembered and performed rather than to any belief about the quantity three.