Both the label and the UUID are part of the file system, not the partition or something inherent to the disk.
The question I linked to and that you deemed not helpful, explained that the UUID is stored in the superblock, which is a part of the file system and fully contained within the partition.
So, if you do a
dd if=/dev/sda1 of=/dev/sdb1, both
sdb1 will have the same label and UUID.
Edit: More details.
Google exist and it can tell you what a superblock is. This site is for professionals and I expect you to be able to get that kind of information yourself.
An UUID is quasi-guaranteed to be unique at creation time. There is no explicit or implicit guarantee that it stays unique if it applied to an object and that object is cloned (as it is the case with a
dd'ed file system. To make it explicit: If you clone a file system, the UUID will not be unique anymore as it is a part of the cloned filesystem.
A MAC can't be considered unique for a long time now. You can change a MAC at will in many/most cases. The same is true with any aspect of a virtual machine.