TL;DR: To access newer versions of mysql/mariadb as the root user, after a new install, you need to be in a root shell (ie sudo mysql -u root, or mysql -u root inside a shell started by su - or sudo -i first)
Having just done the same upgrade, on Ubuntu, I had the same issue.
What was odd was that
sudo /usr/bin/mysql_secure_installation
Would accept my password, and allow me to set it, but I couldn’t log in as root via the mysql client
I had to start mariadb with
sudo mysqld_safe --skip-grant-tables
to get access as root, whilst all the other users could still access fine.
Looking at the mysql.user table I noticed for root the plugin column is set to unix_socket whereas all other users it is set to ‘mysql_native_password’. A quick look at this page: https://mariadb.com/kb/en/mariadb/unix_socket-authentication-plugin/ explains that the Unix Socket enables logging in by matching uid of the process running the client with that of the user in the mysql.user table. In other words to access mariadb as root you have to be logged in as root.
Sure enough restarting my mariadb daemon with authentication required I can login as root with
sudo mysql -u root -p
or
sudo su - mysql -u root -p
Having done this I thought about how to access without having to do the sudo, which is just a matter of running these mysql queries
GRANT ALL PRIVILEGES on *.* to 'root'@'localhost' IDENTIFIED BY '<password>'; FLUSH PRIVILEGES;
(replacing <password> with your desired mysql root password). This enabled password logins for the root user.
Alternatively running the mysql query:
UPDATE mysql.user SET plugin = 'mysql_native_password' WHERE user = 'root' AND plugin = 'unix_socket'; FLUSH PRIVILEGES;
Will change the root account to use password login without changing the password, but this may leave you with a mysql/mariadb install with no root password on it.
After either of these you need to restarting mysql/mariadb:
sudo service mysql restart
And voila I had access from my personal account via mysql -u root -p
PLEASE NOTE THAT DOING THIS IS REDUCING SECURITY Presumably the MariaDB developers have opted to have root access work like this for a good reason.
Thinking about it I’m quite happy to have to sudo mysql -u root -p so I’m switching back to that, but I thought I’d post my solution as I couldn’t find one elsewhere.