Yakuza cut their left pinkie finger. Shanks lost his left 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.
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.
And actually, even earlier than the Yakuza, finger-cutting comes from medieval japanese gamblers. Losing a finger was seen as a way to resolve a bet if you didn’t have another way to pay. This really makes that “I bet it on the Next Generation.” line come across a different way to me.
It’s not the focus of this and I’ve 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.
The Marines don’t call him “captain” by the way, they call him “ogashira/大頭”, or “big boss” like he’s the head of a criminal organization. His crew just uses “kashira/頭”, a more informal “boss”. 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. Rather than the group leader, he’s the head of the family. Same idea, different phrasing.
We also have the father/son cups sake ritual. The Yakuza use the sakazuki cups to cement organization structure and loyalty. The Straw Hat Grand Fleet start calling Luffy “ogashira” after it, even though they don’t do the “real” exchange of cups.
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. Also, Shanks is from the West Blue, where the best sake comes from, and he’s always got a supply. Coincidentally, the West Blue is overrun by underworld mafia figures under the Five Families of the West, and any importing of booze probably goes through them, so Shanks probably has ties to and influence with the Underworld.
How far does it actually go? Well, we know the Underworld business goes all the way to the top of the Government. We know Shanks can just walk into the Holy Land of Mary Geoise and discuss whatever he wants with the Five Elders. We know the Underworld is acting as a go-between for the World Governemnt, the Yonko, and other weapons dealers and criminals across the world.