
Yes I read the whole novel, and they are indeed not brothers. Plus the MC already stated in the newest chapter that he know that Joffrey wasn't the real child of the emperor.
Spoilers (only for those who like to spoil things for themselves, proceed at your own warning)
Joffrey is the actual son of another man back in the queen country, and the emperor knows this that's why he hates Joffrey and is hard on him.
Later in the novel, the queen will kill the Edward's mother and the king will kill the queen later on but will spare the Joffrey at the request - more like threatening- of Edward.
Edward currently cares for Joffrey because he's the only one who gave him attention, warmth and care. When Edward will know that Joffrey isn't related to him he'll pursue him.
In the end, Joffrey became the king of his mother's home country and Edward will marry him.
I didn't want to believe it, but damn, his brother gonna be his love interest