Back in those days the car manufacturers aimed to have more than one supplier of every component, in case there was a supply interruption from one of them. All parts, regardless of vendor, had to be interchangeable, so that the production line logistics would be as simple as possible. Thus they did, indeed, have multiple sources, and so long as the part had the same part number (which was not affected by who supplied it), any combination of them could be used in production. Your Mini is an example of a car built that way.
That system eventually changed, for practical reasons. Often one supplier's component was better than the other's, so having multiple suppliers meant using inferior components on some cars. Testing all combinations of parts on all car models made life very complicated for the car maker's engineering department. Essentially, the system amounted to making a worse product in order to have a more orderly assembly line logistics procedure. The Japanese never did it that way, and eventually, the better results they were achieving had an effect on all of the other manufacturers. Single vendors were chosen for many components, and where there was more than one vendor, the parts were not necessarily interchangeable as parts, but only as systems. Lots of vendors, who had proved themselves incapable of making world-class components, simply went away. The manufacturing logistics people were simply told to suck it up. Generally speaking, the cars got a whole lot better as a result.
