Database backup questions When we back up a database or log, what steps determine the name of the media set? Is it WITH MEDIANAME clause when we format the media? Is this information kept inside the header?
Or is this for naming a particular media (such as a particular tape)? Even though I don't see the point of naming the individual tape.

If I add a subsequent backup sets to the same media set, is it possible to change the media set name in anyway?

Thanks
According to Books Online, a media set is an:


  [O]rdered collection of backup media, tapes or disk files, to which
  one or more backup operations have written using a fixed type and
  number of backup devices.


This comprises an ordered set of backup media, which can be tape, disk files, Azure Blob Storage, etc., but (and this is important) not a combination of two or more of these.

So if you create a media set, your backup media in that media set must be of the same kind.

On to your question about the name of the media set. According to the same article (emphasis added):


  Every volume of backup media (disk file or tape) contains a media
  header that is created when [sic] by the first backup operation that uses
  the tape (or disk). That header remains intact until the media is
  reformatted.


The media set name is assigned by the WITH MEDIANAME keyword, as you surmise. It is not possible to change the name once you have created the first backup media in the set, because it is set by the first backup operation.

You will need to explicitly format the media set if you wish to change its name.