There are various sorts of reinstallation OS disks, however, they are not universal. Dell, for example, has it's own sort of OS reinstallation disks that contain codes specific for that computer. And yes, this is a reinstallation disk and not a restoration disk. Other companies may provide restoration discs, which restore the computer back to factory settings. In some instances, the owner is provided instructions for creating them when they purchase the computer; however, many owners do not do so. The most common method these days, is to have a recovery partition on the computer. That is used to restore the computer back to factory settings.
In a situation in which a recovery partition doesn't exist or has been ruined, and the owner doesn't have reinstallation disks, the owner needs to contact the company that built the computer and inform them of the situation and ask for reinstallation disks. The company will send them out. There is a cost for the disks, but you are not purchasing a new OS or key.
What I've written here concerns prebuilt computers. Custom built is somewhat different. In any case, any set of discs used to reinstall an OS must have a key associated with it.
Note, that there are many other things that need to be installed when reformatting and reinstalling: chipset drivers, keyboard stuff, mouse stuff, and a myriad of other things, so it's not just the OS and the motherboard.