Houshi have a different understanding of family than we do. For them, "family" and "business" are the same thing. For example, the daughter never calls her parents by mom and dad while in the hotel, but by their titles (Okami and President)
[edit: got a bit confused who did what with grandma]
The title is correct because the title doesn't say "same bloodline for 1,300 years".
If you are arguing otherwise, feel free to tell all the adopted people out there that they really are not part of their respective families, and those who adopted others that the people they've adopted are just strangers in their homes.
"Adopted" when you're 40 and happen to be their top manager? Not so much.
Otherwise, the title of Imperator/Augustus went to male blood heirs or usurping Generals (in one case, to a wealthy man who bought off the Praetorian Guard).
"Some males are now joining dating sites specifically made for men looking to become Mukoyōshi, and be adopted by families who need a male successor for their businesses."
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...
But the act of passing the business down to non-blood related adults for 1300 years without going belly up is amazing.