It's a common misconception that MINI is a manufacturer, but it's not. BMW is the manufacturer, MINI is a badge. A German brand with a British badge does not make it a British car.
When it comes to Bugatti, they're manufactured by Bugatti Automobiles S.A.S., not by Volkswagen.
When it comes to MINI, they are manufactured by BMW. Not by MINI, because MINI does not exist as a company.
Bugatti S.A.S. is owned by Volkswagen, but it's not Volkswagen who manufactures the cars. That is the difference.
MINI is owned by BMW, but it's also BMW who manufactures the cars.
Bugatti is a company, MINI is a badge.