How to assign post to category

You have to register built-in “category” taxonomy for the “Activity” post type:

<?php
function add_categories_to_activities()
{
    register_taxonomy_for_object_type( 'category', 'activity' );
}

add_action( 'init', 'add_categories_to_activities' );

See register_taxonomy_for_object_type() in the Code Reference.