Display sub-taxonomies based on SELECTED parent-taxonomy

Here’s a fully functional snippet for doing this.

• I added the correct WordPress table stylings
• This requires adding my smyles_get_taxonomy_hierarchy to add children of taxonomy to the term object
• This hides the child dropdown if no children exist
• This handles everything (and saves) based on TERM ID not the NAME .. you should always use term ID as if you decide to change the wording/slug later on you will have all kinds of issues

Seems code syntax highlighter on here can’t handle mix of HTML/JS/PHP, so here’s a GitHub Gist of working code:
https://gist.github.com/tripflex/527dd82db1d1f3bf82f761486fcc3303

Breakdown:

You first need to include my function to generate taxonomy array with children in that term object:

if ( ! function_exists( 'smyles_get_taxonomy_hierarchy' ) ) {
    /**
     * Recursively get taxonomy and its children
     *
     * @param string $taxonomy
     * @param int    $parent Parent term ID (0 for top level)
     * @param array  $args   Array of arguments to pass to get_terms (to override default)
     *
     * @return array
     */
    function smyles_get_taxonomy_hierarchy( $taxonomy, $parent = 0, $args = array( 'hide_empty' => false ) ) {

        $defaults = array(
            'parent'     => $parent,
            'hide_empty' => false
        );
        $r        = wp_parse_args( $args, $defaults );
        // get all direct decendants of the $parent
        $terms = get_terms( $taxonomy, $r );
        // prepare a new array.  these are the children of $parent
        // we'll ultimately copy all the $terms into this new array, but only after they
        // find their own children
        $children = array();
        // go through all the direct decendants of $parent, and gather their children
        foreach ( $terms as $term ) {
            // recurse to get the direct decendants of "this" term
            $term->children = smyles_get_taxonomy_hierarchy( $taxonomy, $term->term_id );
            // add the term to our new array
            $children[ $term->term_id ] = $term;
        }

        // send the results back to the caller
        return $children;
    }
}

jQuery/JavaScript code handling:

jQuery( function($){
    // slcustom_categories var should be available here
    $('#parent_category').change( function(e){
        var child_cat_select = $( '#child_category' );
        var term_id = $(this).val();

        console.log( 'Parent Category Changed', term_id );

        // Remove any existing
        child_cat_select.find( 'option' ).remove();

        // Loop through children and add to children dropdown
        if( slcustom_categories && slcustom_categories[ term_id ] && slcustom_categories[ term_id ]['children'] ){
            console.log( 'Adding Children', slcustom_categories[ term_id ]['children'] );
            $.each( slcustom_categories[term_id]['children'], function( i, v ){
                console.log( 'Adding Child: ', v );
                child_cat_select.append( '<option value="' + v['term_id'] + '">' + v[ 'name' ] + '</option>');
            });

            // Show if child cats
            $( '#child_category_row' ).show();
        } else {
            // Hide if no child cats
            $( '#child_category_row' ).hide();
        }
    });

    // Trigger change on initial page load to load child categories
    $('#parent_category').change();
});

Function to output fields:

function slcustom_user_profile_fields( $user ){

    $categories = smyles_get_taxonomy_hierarchy( 'project_category' );
    $parent_category = $user->parent_category;
    $child_category = $user->child_category;
//  $parent_category = 52; // used for testing
//  $child_category = 82; // used for testing
    $parent_has_children = ! empty( $parent_category ) && $categories[ $parent_category ] && ! empty( $categories[ $parent_category ]->children );

    // Creative way to use wp_localize_script which creates a JS variable from array
    // You should actually change this to load your JavaScript file and move JS below to that file
    wp_register_script( 'slcustom_user_profile_fields', '' );
    wp_localize_script( 'slcustom_user_profile_fields', 'slcustom_categories', $categories );
    wp_enqueue_script( 'slcustom_user_profile_fields' );
    ?>
    <h1 id="temppp">Select a parent taxonomy</h1>
    <table class="form-table">
        <tbody>
            <tr>
                <th>
                    Parent
                </th>
                <td>
                    <select name="parent_category" id="parent_category">
                    <?php
                        foreach( (array) $categories as $term_id => $cat ){
                            ?>
                            <option value="<?php echo esc_attr( $term_id ) ?>"<?php echo selected( $parent_category, $term_id ); ?>><?php echo $cat->name; ?></option>
                            <?php
                        }
                    ?>
                    </select>
                </td>
            </tr>
            <tr id="child_category_row" style="<?php if( ! $parent_has_children ){ echo 'display: none;'; }?>">
                <th>
                    Child
                </th>
                <td>
                    <select name="child_category" id="child_category">
                        <?php
                            if( $parent_has_children ){
                                foreach( (array) $categories[$parent_category]->children as $c_term_id => $child ){
                                    ?>
                                    <option value="<?php echo esc_attr( $c_term_id ) ?>"<?php echo selected( $child_category, $c_term_id ); ?>><?php echo $child->name; ?></option>
                                    <?php
                                }
                            }
                        ?>
                    </select>
                </td>
            </tr>
        </tbody>
    </table>
    <?php
}