[@mikedean Participant 7 years, 1 month ago]

A year later but here it is.

This assumes a drop down select menu has been added to the form-topic.php template. wp_dropdown_categories() is used to make a <select> element for all the terms in the categories taxonomy so users can assign the topic to a term.
https://codex.wordpress.org/Function_Reference/wp_dropdown_categories

Then, the topic is assigned to the term they selected by hooking into the new topic process.

/**
 * Assign a taxonomy term to a topic after it has been selected in the front end (using a <select> element).
 *
 * @param Int           $topic_id               The id for the current topic being created.
 * @param Int           $forum_id               The id for the current forum the topic is being created in.
 * @param Int           $anonymous_data         ??
 * @param Int           $topic_author           The id of the user who created the topic.
 *
 * @see https://codex.wordpress.org/Function_Reference/wp_dropdown_categories
 */

//Update the taxonomy terms when a new topic is created or edited.
add_action( 'bbp_new_topic', 'myprefix_assign_category_to_post', 20, 4 );

function myprefix_assign_category_to_post( $topic_id, $forum_id, $anonymous_data, $topic_author ) {
        
        //Get the category submitted by the user
        $taxonomy_name = 'category';                    //the slug, or machine name, or the taxonomy
        $term_id = intval( $_POST['cat'] );             //Get value from the <select> element as an interger

        //@TODO error checking if term or taxonomy does not exists (returns false if so)
        $term_object = get_term_by( 'id', $term_id, $taxonomy_name );

        //Replace the existing subject with the selected one
        wp_set_post_terms( $topic_id, array( $term_id ), $taxonomy_name, false );
}