For some reason, by default, WordPress only includes the “Get Shortlink” button when editing posts; not when editing any kind of custom post type or when editing pages. Honestly, I’m not sure why, since pages and custom post types all use the same basic short URL as standard posts (example.com/?p=[post_id]).
The solution is simple, though. You just need to hook into the get_shortlink filter. Following is a simple function that will help you add the button to all “publicly_queryable” post types in your theme.