Yes, He's adopted.
From looks it's guaranteed that he's adopted and in one of the conversations, Christine (Lady Bird) implies that her brother gets good grades in school because of his Race, then he calls her a f**king racist.
This article explains in detail about Unacknowledged Adoptive Relationships in the Film Ladybird.
Miguel is a handsome man of color, probably mixed race, and in his twenties. We learn only a bit about this mild mannered young man, but he seems to be on his way to success in the work world. He appears caring toward Ladybird, helping her get a job, but she seems uninterested in him.
Everyone I know who has seen the film–both those involved in adoption and not–have been puzzled by his place in the family, concluding he must be adopted. Then, Ladybird, who is white, might be adopted too. The many ways in which she is different from her mother could be the result of adoption and could explain the mother’s inability to effectively parent such a daughter. An adopted daughter, with no ties to her birth family, might have an extra incentive to change her name.
And also at the end of the movie, in one of the Marion's letters to Christine, she mentions that Christine was a happy accident.
close-ups on Marion's handwritten letters reveal some interesting tidbits, such as the fact that Christine was a happy accident. (The exact wording implies fertility problems, which also neatly explains the presence of the clearly-adopted Miguel.)