Context is important here, but no, they are usually not the same. "It's my honor" - would be used for an opportunity that you're honored to be offered. "It is my honor to present this award." "On my honor" is used to verify truth or goodness. Essentially, you are swearing on your honor (your respectability and worthiness) that you are telling the truth. "Did he really do that?!" "On my honor." Would be a way of emphasizing seriousness and truth in what you are saying. "My honor" is way too open-ended to really say. But when used as a form of address, which I think is what you mean, is used to address a king or lord (but not like a president or prime minister. It has a distinctly feudal connotation.) "Your honor" is still used in modern day to address a court judge.