show menu to specific membership type or role

You can do:

in_array('administrator', $GLOBALS['current_user']->roles)

That checks whether current user administrator role. If user is not logged in, the roles array will be empty so the menu item doesn’t show.

But what happen if “logged_in_user” share the site url or link.

The Menu And its Items visible to others without logged in

How to prevent it

This plugin only deals with the display of menu items. If you wish to protect your pages you need another plugin, for example: https://www.ads-software.com/plugins/members/.