Ubuntu Server 14.04: No login with new MariaDB users

Try running mysql_secure_installation - improve MySQL installation security, as Michael mention, it almost sounds like you already have mysql database with something other than fresh install.

You can also try stop mariadb.service delete mysql database and then try to give it another try with mysql_install_db - initialize MySQL data directory.

Some Debian based distributions have started to use the UNIX socket authentication plugin for the root user. This means that the operating system account is linked to the database root account and you can only access the database if you have superuser privileges. To my knowledge, this is done to make the initial installation more secure.

To verify that it is indeed the UNIX socket authentication plugin that is causing this, you can log in as another user and try to connect to the database with mysql -uroot. If it fails, you've just confirmed that the database uses the UNIX socket authentication plugin.

By default, the mysql client will use a local UNIX domain socket to connect to a server. To use a network connection, give the loopback address as the hostname with -h 127.0.0.1. This should allow you to connect as the root user with an explicit password.