MYSQL refuses to set password for new user
I am going to take a very wild guess on this one
My guess is that you upgraded to MySQL 5.7 but did not upgrade the the mysql schema.
I wrote about the mysql.user table having a different number of columns per version:

Oct 10, 2014 : MySQL service stops after trying to grant privileges to a user
Aug 07, 2013 : MySQL - ERROR 1045 (28000): Access denied for user : permission issue
May 24, 2013 : How often should MySQL (stock, Percona, etc.) be upgraded?
May 01, 2013 : How often should MySQL (stock, Percona, etc.) be upgraded?
Apr 12, 2012 : Cannot GRANT privileges as root (I actually show the differences in the column layouts in MySQL 5.6, 5.5, 5.1, 5.0, 4.x)

Now for the shocker
Did you know the following about mysql.user in MySQL 5.7 ?

There are 45 columns
There is no longer any column named password
The column authentication_string stores the password now
I mentioned this Oct 28, 2015 : Install MySQL for Windows from .zip and reset root password

At this point, you are probably asking : Why in the world did create user 'testing'@'localhost' identified by 'password'; work ?
Look in my Apr 12, 2012 post. Please note the the column layout for mysql.user. The column authentication_string column appears in MySQL 5.6 and MySQL 5.5.
Please run this query
select Host, User, password_expired,Password,authentication_string from mysql.user;

You should see the encrypted password in the authentication_string column
What makes me believe this to be the case was the clue you provided
mysql&gt; SHOW GRANTS FOR 'root'@'localhost';
+---------------------------------------------------------------------+
| Grants for root@localhost                                           |
+---------------------------------------------------------------------+
| GRANT ALL PRIVILEGES ON *.* TO 'root'@'localhost' WITH GRANT OPTION |
| GRANT PROXY ON ''@'' TO 'root'@'localhost' WITH GRANT OPTION        |
+---------------------------------------------------------------------+
2 rows in set (0.00 sec)

The command SHOW GRANTS; no longer shows the encrypted password in MySQL 5.7. I ran in to this issue even with pt-show-grants.
What should you do ?
Perhaps you should run mysqld --upgrade to fix the column layout. It looks like this in 5.7:
mysql&gt; desc mysql.user;
+------------------------+-----------------------------------+------+-----+-----------------------+-------+
| Field                  | Type                              | Null | Key | Default               | Extra |
+------------------------+-----------------------------------+------+-----+-----------------------+-------+
| Host                   | char(60)                          | NO   | PRI |                       |       |
| User                   | char(32)                          | NO   | PRI |                       |       |
| Select_priv            | enum('N','Y')                     | NO   |     | N                     |       |
| Insert_priv            | enum('N','Y')                     | NO   |     | N                     |       |
| Update_priv            | enum('N','Y')                     | NO   |     | N                     |       |
| Delete_priv            | enum('N','Y')                     | NO   |     | N                     |       |
| Create_priv            | enum('N','Y')                     | NO   |     | N                     |       |
| Drop_priv              | enum('N','Y')                     | NO   |     | N                     |       |
| Reload_priv            | enum('N','Y')                     | NO   |     | N                     |       |
| Shutdown_priv          | enum('N','Y')                     | NO   |     | N                     |       |
| Process_priv           | enum('N','Y')                     | NO   |     | N                     |       |
| File_priv              | enum('N','Y')                     | NO   |     | N                     |       |
| Grant_priv             | enum('N','Y')                     | NO   |     | N                     |       |
| References_priv        | enum('N','Y')                     | NO   |     | N                     |       |
| Index_priv             | enum('N','Y')                     | NO   |     | N                     |       |
| Alter_priv             | enum('N','Y')                     | NO   |     | N                     |       |
| Show_db_priv           | enum('N','Y')                     | NO   |     | N                     |       |
| Super_priv             | enum('N','Y')                     | NO   |     | N                     |       |
| Create_tmp_table_priv  | enum('N','Y')                     | NO   |     | N                     |       |
| Lock_tables_priv       | enum('N','Y')                     | NO   |     | N                     |       |
| Execute_priv           | enum('N','Y')                     | NO   |     | N                     |       |
| Repl_slave_priv        | enum('N','Y')                     | NO   |     | N                     |       |
| Repl_client_priv       | enum('N','Y')                     | NO   |     | N                     |       |
| Create_view_priv       | enum('N','Y')                     | NO   |     | N                     |       |
| Show_view_priv         | enum('N','Y')                     | NO   |     | N                     |       |
| Create_routine_priv    | enum('N','Y')                     | NO   |     | N                     |       |
| Alter_routine_priv     | enum('N','Y')                     | NO   |     | N                     |       |
| Create_user_priv       | enum('N','Y')                     | NO   |     | N                     |       |
| Event_priv             | enum('N','Y')                     | NO   |     | N                     |       |
| Trigger_priv           | enum('N','Y')                     | NO   |     | N                     |       |
| Create_tablespace_priv | enum('N','Y')                     | NO   |     | N                     |       |
| ssl_type               | enum('','ANY','X509','SPECIFIED') | NO   |     |                       |       |
| ssl_cipher             | blob                              | NO   |     | NULL                  |       |
| x509_issuer            | blob                              | NO   |     | NULL                  |       |
| x509_subject           | blob                              | NO   |     | NULL                  |       |
| max_questions          | int(11) unsigned                  | NO   |     | 0                     |       |
| max_updates            | int(11) unsigned                  | NO   |     | 0                     |       |
| max_connections        | int(11) unsigned                  | NO   |     | 0                     |       |
| max_user_connections   | int(11) unsigned                  | NO   |     | 0                     |       |
| plugin                 | char(64)                          | NO   |     | mysql_native_password |       |
| authentication_string  | text                              | YES  |     | NULL                  |       |
| password_expired       | enum('N','Y')                     | NO   |     | N                     |       |
| password_last_changed  | timestamp                         | YES  |     | NULL                  |       |
| password_lifetime      | smallint(5) unsigned              | YES  |     | NULL                  |       |
| account_locked         | enum('N','Y')                     | NO   |     | N                     |       |
+------------------------+-----------------------------------+------+-----+-----------------------+-------+
45 rows in set (0.00 sec)

mysql&gt;