Considering they're the same age, I would actually like if they were childhood friends, the only kid Yamato had around to play with, yet they became distant as Jack grew within the crew. Oda could even play with some subplot about Jack being like a son trying to impress his dad, hence his appearance and his willingness to take care of things that are important to Kaido. This would add another layer to Kaido's moment protecting Jack from the minks, too.