Regarding the “Brother” and “exchange sake cups” thing…

In the yakuza, two people who make pledges to each other over sake cups (drinking together and exchanging cups with each other) are affirming and tightening their “brothership”, becoming like “sworn brothers”.

Saejima and Majima also have this sworn brothership with each other.

Ryu ga Gotoku 5 Cutscenes Part 15/38