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.
My adopted Children when Adopted (10, 20, 22) They were in my house at 7, 11, 15 from the same mother. LONG story but I ended up adopted the two older ones when they were ready which wasn't till they were adults.
Nobody is arguing that adoption per se isn't the same as "the same family". The practice being described here is that the adoption is specifically adopting someone _because_ they're going to be the next owner of the business. I'm sure you can see that there's a salient difference between your case of adopting to build a family vs the case here of a 65 year old guy adopting his 50-year old successor as part of a tradition of that business.
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.