UPDATE with Nested Query
If it is MySQL, you can do this with a simple join:

UPDATE music AS m
  JOIN albumList AS a
    ON m.songalbum = a.album 
SET m.albumID = a.albumID 
WHERE m.albumID IS NULL ; 




You could use a correlated subquery for this (similar to your 2nd attempt) but I wouldn't suggest it for MySQL. Less efficient and more complicated as it needs an additional condition in the WHERE clause to protect from updating music.albumID with null, for rows that have no related album:

UPDATE music AS m
SET albumID = (SELECT albumID FROM albumList AS a WHERE m.songalbum = a.album) 
WHERE albumID IS NULL
  AND EXISTS (SELECT 1 FROM albumList AS a WHERE m.songalbum = a.album) ;