
No, they are not half brother.
In chapter 5, Miyauchi said that his mother is a servant have relationship with the misstress of the family. She was later discarded because the misstress going to marry. When her mother die, a butler from that said family came and train Miya to butler and then send him to Saionji Family where he later meet Aoi. If I not wrong that picture show that the butler that train Miyauchi is Jirou father which is butler from the Saionji family.

Yeah, they are half-brothers. Jirous's mother was the "father" of Miyauchi. Jirou's mother had Miyauchi with her family's servant. But she had to marry Jirou's father, so she left the servant (Miyauch's mother) and married Jirou's father and had Jirou as the result of their union. When Miyauchi's mother died, Kudo's father took Miyauchi under his care and trained him to be Aoi's butler. Simple as that... hope that was clear (づ ̄ ³ ̄)づ
do the last few pages mean that jirou and miyauchi are half-brothers???