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 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 1) again.