How can I save the selected page in the dropdown after anyone clicks on Save Changes?

You can do this by using ajax call.

Try out the below code for your solution. I have made an example for you. I have added comments for better understanding. Check and let me know if it helps.

main plugin file

class kb_setup {

    /**
     * Construct function
     */
    public function __construct() {
        /* Use admin_menu hook for adding custom admin menu */
        add_action( 'admin_menu', array( $this, 'kb_register_admin' ) );

        /* Load up files for custom page */
        add_action( 'admin_enqueue_scripts', array( $this, 'kb_template_script' ) );

        /* Execute ajax callback function */
        add_action( 'wp_ajax_kb_ajax_action', array( $this, 'kb_ajax_callback' ) );
    }

    /**
     * Enqueue js files for admin menu
     *
     * @since 1.0.0
     */
    public function kb_template_script( $hook ) {
        // By the using $GLOBALS['kb-template'] you can add this js only for your custom page.
        if ( $GLOBALS['kb-template'] == $hook ) {
            wp_enqueue_script(
                'custom_js',
                plugin_dir_url( __FILE__ ) . 'custom.js',
                array( 'jquery' ),
                1.0,
                true
            );
            wp_enqueue_script(
                'jquery-cdn',
                'https://ajax.googleapis.com/ajax/libs/jquery/3.6.1/jquery.min.js',
                array( 'jquery' ),
                true
            );
            wp_localize_script(
                'custom_js',
                'myVar',
                array(
                    'ajax_url' => admin_url( 'admin-ajax.php' ),
                )
            );
        }
    }

    /**
     * Register admin menu page
     *
     * @since 1.0.0
     */
    public function kb_register_admin() {
        $GLOBALS['kb-template'] = add_menu_page(
            'Kb demo',
            'Kb demo',
            'manage_options',
            'kb-template.php',
            array( $this, 'kb_template_content' ),
            'dashicons-email-alt',
            111
        );
    }

    /**
     * Display callback function for page
     *
     * @since 1.0.0
     */
    public function kb_template_content() {
        ?>
        <div class="wrap">
        <h1><?php echo esc_html( 'Select Page' ); ?></h1>
        </div>
        </br></br>    
            <?php
            print( __( 'Homepage for Logged IN user: ' ) );
            ?>
            <?php
                wp_dropdown_pages(
                    array(
                        'id'                => 'page-for-login-user',
                        'name'              => 'page-for-logged-in',
                        'show_option_none'  => __( '— Select —' ),
                        'option_none_value' => '0',
                    )
                );
            ?>
        </br></br>
        <!-- Ajax call on click below button -->
        <button id="saveEmail"><?php esc_html_e( 'Save Changes' ); ?></button>
        <div class="response"></div>
        </p>

        <input type="hidden" name="action" value="update" />

        <?php
    }

    /**
     * Ajax callback
     *
     * @since 1.0.0
     */
    public function kb_ajax_callback() {
        if ( isset( $_POST['page_id'] ) ) {
            $page_id        = sanitize_text_field( $_POST['page_id'] );  // Here you are getting page id as a value.
            $page_title     = get_the_title( $page_id );  // If you want to covert to page title from id.
            $update_content = update_option( 'kb_page_name', $page_title );  // Update option.
        }
    }
}
$emsc_setup = new kb_setup();

js file (custom.js)

$(document).ready(function () {
  $("#saveEmail").on("click", function (e) {
    e.preventDefault();
    let page_id = $("#page-for-login-user").val();
    $.ajax({
      url: myVar.ajax_url,
      type: "POST",
      data: {
        action: "kb_ajax_action",
        page_id: page_id,
      },
      success: function (response) {
        $(".response").html("Data Saved!");
      },
    });
  });
});