He has the right. His father was legitimately married to his brother's mother. The Hugh family probably only had daughters and no sons but it's a Duchy and so it needs to continue. Having a man marry in and take over the family was normal. So his father became the rightful heir of the family name, so when his brother's mother died and eventually got into another relationship and Zachary was born, Zachary is still a rightful heir but just not a Hugh by blood but by name. His brother is being vindictive for no reason other than feeling resentment bc he felt like Zachary stole everything from him when he didn't. Zachary being born isn't his fault and basically his brother thinks it is.
Wait a damn minute so if Zachary has no right to inherit or rule their family(the one they grew up with) why did his brother kick him out at the beginning of the story since his brother is the only heir here since the whole thing belonged to his mother and her family and Zachary has no relation to it