You're looking in the wrong place, running mysql as (system) user mysql won't solve anything.
mysql is a client program, the authentication to the server, run by mysqld, is made on the connection (tcpdump port 3306 to make it clear).
Therefore, you can launch mysql with any "system" user, like:
$ mysql -u root -p localhost
Give the admin ("root") password when prompted, and then you should see this:
Welcome to the MySQL monitor. Commands end with ; or \g.
Your MySQL connection id is 240
Server version: 5.5.38-0ubuntu0.14.04.1 (Ubuntu)
Copyright (c) 2000, 2014, Oracle and/or its affiliates. All rights reserved.
Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.
Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.
mysql>
...and there you go!