What is the significance of phpMyAdmin's configuration storage and what should I know about it?
There was a broken link in that wiki page (which I've just fixed), but it was supposed to take you to http://docs.phpmyadmin.net/en/latest/setup.html#linked-tables

Basically, they're additional features including bookmarked queries, comments, SQL-history, tracking mechanism, PDF-generation, transformations, relations, and more. By adding the tables (which you can do manually or phpMyAdmin often can do for you, which you have taken advantage of), you get access to these features. If you don't need the features, then you don't need the tables.

So you're correct that you don't need to manually run the SQL statements now.

The tables start with the pma__ prefix for the same reason WordPress tables start with wp_ and other applications use similar prefixes: in shared hosting situations, users often only get access to one database and several applications need to peacefully coexist. By adding a namespace prefix, you can have gallery_users, wp_users, and gramps_users (I'm making up these table names for the sake of example)