Interesting question! There’s a few of us that work in the industry on here, so no doubt you’ll get some informed answers.
Firstly, it entirely depends on the end product, volume, and OEM building it. I’ll give an example using an engine block (most cars have them):
Low volume (let’s say sub 10,000 vehicles/annum) aren’t going to purchase a foundry, and tool up to manufacture <10k blocks a year, financially it doesn’t work. So they will buy capacity from a supplier (Grainger & Worrall for example). GW as a component supplier, are then contracted to provide ‘x’ number of engine blocks to OEM timing/quality etc.
The reverse of that is a high volume player, with global product presence, component shared across platforms, big boy stuff. This is when you might well find Ford/Volkswagen own a supplier/foundry so they have complete control over the manufacturing chain.
Clearly an engine block is bespoke to individual OEMs. But, what about something like an indicator blub, or a map sensor? Generic components are often much cheaper to be purchased from a supplier, as they have ultimate economies of scale.
But, clearly the manufacturing process is a very small part of the bigger picture. OEMs control every part of the process; right from initial design, development, manufacture, integration, calibration, type approval, assembly etc. That excludes the interesting bits like research, concept development and advanced engineering, or the customer facing roles like dealerships.
So, to answer your question:
OEMs may make some of the raw parts, depends on the OEM, and what the part is. A Bentley seat for example is stitched together on site, clearly they don’t breed and manage cows for the leather directly.
Other businesses often build components of sub-assemblies. The exception to this is probably powertrains, where whole assemblies are purchased from gearbox suppliers (ZF for example, are in Jag, Bentley, BMW, Audi etc)
Regarding sharing across brand, in the scheme of things probably quite a small percentage of parts. A gearbox is a good example, ZF make the 8-speed box popular in any premium car for the last 3 years. OEM engineers integrate and calibrate this. So although your BMW shares the same box with an Audi, it can behave and drive completely differently (brand DNA).
Interesting subject, but very dependent on which component you’re asking about, and which OEM!