When a player buys a relationship in character burning, they get to define the nature of the relationship, the relationship’s attitude toward the character, and the relationship’s importance to the setting.
Outside character burning, all of those decisions are the GM’s to make. A player can exert some control by making a successful Circles test, but they are limited to characters that fall withing the circles described by their lifepaths (or a trait that opens up new circles).
You could write a belief about how the duke is your brother, but even if that’s true, you won’t be able to circle them if you’re a peasant. You don’t have the circles for that. Moreover, regardless of your belief, the GM gets to fully define that character and what the relationship actually is. Writing something as a belief doesn’t make it objectively true. It just means the GM needs to engage you over that belief in some way.
That said, the GM can certainly run with it! The duke could be the character’s brother. It’s up to the GM. If you want to decide what the duke is like, how they feel about you and if they’re actually your brother, buy them as a relationship in character burning. Otherwise, you leave all those decisions in the GM’s hands.
You don’t have to buy your family as relationships, but neither do you have automatic access to them if you don’t. If you want to bring them into a scene, the GM has the option to say ‘yes’ as usual, or to call for a Circles test. Remember, Circles is meant to represent people from your character’s past. That includes family.