In the west the business would get sold and merged. Only in Japan does the adoption make it a family or lineage thing.
No. Every culture worldwide accepts adoption. A child, grandchild, nephew, etc, continuing a business keeps the business in the family. Whether the child is adopted or not, and regardless of the age of adoption.
In Japan, the longevity of the thing has a value of its own, so much so that they’ve invented adult “adoptions”, simply to keep an old business going. The business could be mediocre and barely profitable, but if it’s 300 years old, they’ll find a way to keep that line going.
Maybe they exist, but I’m not aware of any other culture that has taken up the practice of adopting adult males to ensure that a family business remains “family”.
Tge street-level character of a city isn't magically preserved because the acquiring company chose to retain the name. Example: I live near a sushi restaurant that has changed hands six times in my life. Two times they kept the name. One time they kept the facade. Never did they keep the employees. Maybe a token "manager" but the character of the place always shifts.
I like the metaphor because it really does ask the question "how much must one replace to make something indistinguishable from its progenitor?"
Are adult adoptions common at all?
Note how often these adult men are being adopted along with marrying the daughter of the family. I dare say outside of Japan such an arrangement would be considered weird.
How does that work? They become both husband and brother to the same person?
In practice what it means is that the couple adopts the family name of the wife, not the husband, which is the name associated with the family business that he's taking over.
E.g. Fuller's (London brewer) now owned by Japanese conglomerate, Church (Northampton shoes, was still in the family) bought by Prada.