MySQL fails on: mysql "ERROR 1524 (HY000): Plugin 'auth_socket' is not loaded"

My local environment is:

• Fresh Ubuntu 16.04
• With PHP 7
• With installed MySQL 5.7

sudo apt-get install mysql-common mysql-server

When I tried to login to MySQL (via CLI):

mysql -u root -p

I came across an cyclic issue with 3 steps.

1. First was some socket issue

   ERROR 2002 (HY000): Can't connect to local MySQL server through    socket '/var/run/mysqld/mysqld.sock'

   Solution: Restarting PC.

   Which led to another error:

2. With access denied

   ERROR 1698 (28000): Access denied for user 'root'@'localhost'.

   Possible issue? Wrong password for root user!

   Solution: Reset root password with this RackSpace tutorial. With correct password and working socket, there comes last error.

3. Incorrect auth plugin

   mysql "ERROR 1524 (HY000): Plugin 'unix_socket' is not loaded"

   Here I stopped or somehow got to step 1. again.