Minami doesn't show any disrespect towards Tatsuya, despite his standing in the Yotsuba family. If there is any strain in their relationship, it would be from Tatsuya himself, as Minami reminds him of her aunt, who he feels responsible for letting die during the Reminiscence arc. As a result, he tries to avoid having any more contact than necessary.
Still, Tatsuya treats Minami with respect, and encourages her to treat him and Miyuki as equals.
Like most of the Yotsuba family, Minami holds Miyuki in high regard as the most likely candidate to succeed Maya. As a result, she believes that her purpose is to serve Miyuki and to become a proper Guardian as soon as possible. This initially conflicts with the Shiba's lifestyle, as she refused to let Miyuki do any housework for while, until this led to a sort of compromise where Miyuki would be in charge of the meals and tea when Tatsuya was around, as well as Tatsuya's wardrobe, while Minami was in charge of everything else. However, Tatsuya feels that if one gave an opening, the other would immediately seize the opportunity.
Aside from this, Miyuki treats Minami with respect, which Minami is unfamiliar with.
At first, She also did not like the relationship between the siblings. Specifically, Miyuki's "inconveniently severe brother complex". After the revelations in Vol 16, She treats them with even more respect, changing from -nee-sama to -sama.
She is heavily attracted to him and becomes embarrassed in front of him.