Morgan wasn't always a Minotaur. His current state is the result of genetic engineering, which was a secret project by officials in the government in the universe he's from. He was taken prisoner (on bogus charges) and used as a human guinea pig. Andromeda was subject to the same treatment.
Before that Morgan (which is actually his last name. Full name is Odysseus Morgan) and Andromeda were a happy new family with a young daughter (Silver). Then Andromeda was supposedly killed, Silver went missing and Morgan was taken away. He was captive for 5 years, and the change took 1 year to complete. He's been rogue (smuggler/pirate) ever since. It's been 10 years since he was initially captured. Asda (the doctor) is the one who helped rescue him, and most of the people who make up his crew. In his former life Morgan worked as a Trader/hauled cargo around the planets.