MySQL 8.0

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
CREATE TABLE example (
  id INT NOT NULL AUTO_INCREMENT Primary KEY,
  fullname VARCHAR(100) NOT NULL,
  email VARCHAR(100) NOT NULL,
  something VARCHAR(255) NOT NULL
);
INSERT INTO example (id, fullname, email, something) VALUES
  (1, 'Name 1', 'name1@example.com', 'Something 1'),
  (2, 'Name 2', '', 'Something 2'),
  (3, 'Name 1', 'name1@example.com', 'Something 3'),
  (4, 'Name 3', 'name3@example.com', 'Something 4'),
  (5, 'Name 3', 'name3@example.com', 'Something 5'),
  (6, 'Name 4', '', 'Something 6');
SELECT GROUP_CONCAT(DISTINCT fullname), email 
FROM example 
WHERE email <> ''
GROUP BY email 
UNION
SELECT fullname, email 
FROM example 
WHERE email = ''
ORDER BY email;


GROUP_CONCAT(DISTINCT fullname)
email


Name 2


Name 4


Name 1
name1@example.com

Name 3
name3@example.com


SELECT GROUP_CONCAT(DISTINCT fullname), email 
FROM example 
GROUP BY email 


GROUP_CONCAT(DISTINCT fullname)
email


Name 2,Name 4


Name 1
name1@example.com

Name 3
name3@example.com