Danger Mouse was just born without his left eye, or lost it in some mundane way like an accident at home or something.
Subversion of expectations. He doesn't have a tragic or interesting backstory. Danger Mouse doesn't have a real name.
His passport only says “Danger Mouse”, which would imply that. The Danger Babies who are training to be agents already have “Danger (species)” type names, maybe DM was originally one of them or a part of a similar program? Agent 57 isn't Agent 58's biological father.
58 is adopted or 57's stepson, and 57's real species is still unknown. Dr. Augustus P. Crumhorn III was an enemy of Colonel K in his agent days.
Crumhorn III seems to have existed since Crumhorn IV is presumably his son, but Crumhorn III isn't in the show so he was probably active in the past. Colonel K's agent days were placed in the eighties, around the original show's time.