Add category selection to custom post type (plugin)

Question

I picked up someone's old project at the company I work at. They created a plugin to add courses to the website (including dates, location etc), but they didn't add the possibility to select a category for the course.

I want to add this function, which I tried by looking at their other code and copy/pasting it and adapting it so it fits my needs.

It does show up on the right side of the page, but whatever I do, it never remembers the category.

I also adapted the "POST" function lower in the file, but I guess I'm still missing a piece of code that actually activates the category-selection, as it is also not showing up in the overview of all courses:

The previous developer worked with add_meta_box to create the boxes on the side. I created the following:

add_meta_box(
        'course_category_box',          // Unique ID 
        esc_html__( 'Category', 'Category Box' ),       // Title
        'render_course_category_box',       // Callback function
        'course',                   // Admin page (or post type)
        'side',                 // Context
        'default'                   // Priority
    );

Which I called upon by using

<?php
    function render_course_category_box( $object, $box ) {

        wp_nonce_field( basename( __FILE__ ), 'course_category_box_nonce' ); ?>
    <p>
         <?php wp_dropdown_categories( $args ); ?> 
    </p>
<?php } ?>

And also added it to the save_course_meta_boxes function.

if ( !isset( $_POST['course_category_box_nonce'] ) || !wp_verify_nonce( $_POST['course_category_box_nonce'], basename( __FILE__ ) ) )
        return $post_id;

Now, I'm probably doing something that is way too complicated, as Wordpress often has a much easier way to add functions like these.

I hope it's clear enough what my problem is. If not, don't hesitate to ask for more info!