Why wasn't Killer Bee recognized as the 5th Raikage?
I do not believe Kishimoto wrote anything in regards to transition of the 4th to the 5th Raikage, but the selection of the Raikage is not touched on a whole lot and is unclear to us.


  How the Raikage is selected isn't clear; after the Third Raikage died, his son automatically became Fourth Raikage, a decision made many years earlier


It is possible Killer B was nominated for the position, but we may never know. I can say that Killer B has no passion to being a Kage - his passion is music and his goal is to be greatest rapper. If he was nominated, Killer B may have turned it down, like Jiraiya, to pursue his passions


  Despite being responsible for the village's protection, B aspires to be the world's greatest rapper. 


While there have been, and currently are, Kage who are jinchūriki, this does not automatically make them the next in line a Kage position. It is merely tradition for the next jinchūriki to be a relative of the current Kage.


  Because jinchūriki were so important to their village, it was a tradition for a jinchūriki to be a relative of the current Kage, both to strengthen the jinchūriki's loyalties to the village and to display the Kage's might.


Given that Killer B is the (adoptive) brother of the 4th Raikage, he became the jinchūriki for his brother.


  According to Motoi, the reason behind him becoming a jinchūriki was for the sake of his brother, even if it meant being shunned by the villagers