But the act of passing the business down to non-blood related adults for 1300 years without going belly up is amazing.
So if you adopt a child and raise it as your own and that child takes over your business, is it still a "family business? Yes.
But can you say it's still a "family" business if it's gone through the Japanese custom of adult adoption? Arguably yes, because that person was brought into their family as a legitimate heir.
On the other hand you could argue that it's no different than a family business being sold in the western sense, it's just that in ancient Japan you couldn't really buy and sell businesses, they were owned by clans, so this was their mechanism for ensuring business continuity when there was no suitable male heir.
I'm not taking a stance on whether it's still a "real" family business, but it's important to understand that what the Japanese mean by "adoption" in this sense is not at all what westerners mean when they use that word.
Then again, the west took on the practice of putting the founders name on a corporations...