SQL Server backup medianame
If I make full/differential/log backups to disk under full recovery
  model it is important to specify MEDIANAME?


NO, it is not important


  Is it true that MEDIANAME is only for tape backups? If not, when and
  why should I specify MEDIANAME in the backup statement?


No it is not just for tape backups,  you can take local disk backup and specify media name. More over its a name given to particular backup set.For example

BACKUP DATABASE AdventureWorks2012
TO DISK='X:\SQLServerBackups\AdventureWorks1.bak', 
DISK='Y:\SQLServerBackups\AdventureWorks2.bak', 
DISK='Z:\SQLServerBackups\AdventureWorks3.bak'
WITH FORMAT,
   MEDIANAME = 'AdventureWorksStripedSet0',
   MEDIADESCRIPTION = 'Striped media set for AdventureWorks2012 database;
GO


I just took backup of Adventureworks database on my local machine and provided media name below screenshot will show you the result.




  Additional question: msdb.dbo.backupmediaset.media_set_id actually
  means backup file id if backup media is disk, it that true?


No they both are different. Below query can easily show it to you. I used media name we just created above in the backup query

select bm.media_set_id,
bs.backup_set_id,
bm.media_family_count,
bm. name,
bm. description 
from msdb.dbo.backupmediaset  bm
inner join  msdb.dbo.backupset  bs
ON bm.media_set_id=bs.media_set_id
where bm.name='AdventureWorksStripedSet0'


And the output is 



So you can see backup_set_id and media_set_id are different.