From https://jayj.dk/multiple-select-lists-theme-customizer/

multiple_select_class.php

/**
 * Multiple select customize control class.
 */
class Jayj_Customize_Control_Multiple_Select extends WP_Customize_Control {

    /**
     * The type of customize control being rendered.
     */
    public $type = 'multiple-select';

    /**
     * Displays the multiple select on the customize screen.
     */
    public function render_content() {

    if ( empty( $this->choices ) )
        return;
    ?>
        <label>
            <span class="customize-control-title"><?php echo esc_html( $this->label ); ?></span>
            <select <?php $this->link(); ?> multiple="multiple" style="height: 100%;">
                <?php
                    foreach ( $this->choices as $value => $label ) {
                        $selected = ( in_array( $value, $this->value() ) ) ? selected( 1, 1, false ) : '';
                        echo '<option value="' . esc_attr( $value ) . '"' . $selected . '>' . $label . '</option>';
                    }
                ?>
            </select>
        </label>
    <?php }
}