In NeOlympus, the Minotaur is the reincarnation of the same ancient monster.
As part of their efforts to adapt their divine powers to the different cosmology of Elysium (the world which "NeOlympus" takes place in), the Olympians poured the souls of past heroes and slain monsters into the Well of Souls. This act stabilized the magic on this new world.
As a result, the souls of monsters now inhabit the bodies of newborn children and upon reaching their teenage years, those with monster souls undergo changes that transform them into the monster of myth. The Minotaur is one such example.
The current Minotaur is the fourth reincarnation of the beast. Rather than live in exile or confinement as it has in previous lives, the Minotaur became a gladiator in the fighting pits of the Spartan Empire, often given the charge of executing criminals and heretics for sport while entertaining the masses who come to watch such spectacle.
The Minotaur's name is Asterion the fifth.