How to display the category dropdown auto search list when key press?

I need to know what query I have to use on the AJAX to get the output?

In the comments, you said, “I am looking to search only for categories“, so for that, you can simply use get_categories() (or get_terms() for custom taxonomies) and set the search arg to the submitted search keyword:

// Get the categories assigned to ANY posts that matched the search keyword.
$terms = get_categories( array(
    'search' => sanitize_text_field( $_REQUEST['universalSearchlist'] ),
) );

// Then display the categories or do whatever is necessary. (I used 'echo' for testing.)
foreach ( $terms as $term ) {
    $cat_link = get_category_link( $term );
    echo '<a href="' . esc_url( $cat_link ) . '">' . esc_html( $term->name ) . '</a><br>';
}

/* Or using get_terms():
$terms = get_terms( array(
    'search'   => sanitize_text_field( $_REQUEST['universalSearchlist'] ),
    'taxonomy' => 'your_taxonomy',
) );

if ( ! is_wp_error( $terms ) ) {
    foreach ( $terms as $term ) {
        ... your code.
    }
}
*/

So there’s no need for WP_Query unless if you wanted only the categories assigned to posts that matched the search keyword.

Additional Notes

1. Actually, I suggest you to use the REST API for searching the categories — e.g. you’d make your AJAX requests to example.com/wp-json/wp/v2/categories?search=<keyword>, but you can ask another question for that.

2. In your universalSearch() function, you should call wp_die() and there’s no need to return anything (the wp_ajax_ hooks are action hooks, not filter hooks).

3. You should really enqueue your scripts (.js) using wp_enqueue_script() and you should just use the default jQuery script that came bundled with WordPress.

4. Your AJAX JS is not actually sending the search keyword to the server, so in addition to the action parameter, you should add universalSearchlist: $( this ).val() to the second parameter for $.post().