I imagine Shanks and his crew did the same father/son cups ritual as any other yakuza family when forming the red hair pirates, considering their deferment to his rank and level of respect. Kyoshiro's family in Wano use it to swear loyalty, so it even has explicit ties to the yakuza within the world of One Piece. the grand fleet start calling Luffy 'ogashira' after it, even though they don't do the 'real' exchange of cups. the yakuza use the sakazuki cups to cement organization structure and loyalty. We also have the father/son cups sake ritual. rather than the group leader, hes the head of the family. the yakuza organization structure IRL places the 'wakagashira'/'young boss' right under the oyabun/kumicho, so shanks' title is alluding to the yakuza without being one of their real titles. His crew just uses 'kashira/頭', a more informal 'boss'. like he's the head of a criminal organization. The marines don't call him 'captain' by the way, they call him 'ogashira/大頭', or 'big boss'.

It's not the focus of this and ive posted it before, but in my opinion, Shanks losing his arm was an intentional act to cement his influence over Luffy, because Shanks could see what Luffy's future might hold.

This really makes that 'I bet it on the next generation.' line come across a different way to me. Losing a finger was seen as a way to resolve a bet if you didn't have another way to pay. And actually, even earlier than the yakuza, finger-cutting comes from medieval japanese gamblers. We know Shanks is a swordsman and lost at least some level of ability in this skill because of Mihawk refusing to duel, and from how he uses it in flashbacks, his left arm was his dominant arm. Traditionally, like from medieval times, the finger-cutting is meant to cripple your swordsmanship, as the pinky finger grips the handle the most tightly.

Giberson the hider