Now, whenever changes are made to the app/map itself, these changes will reflect in the new item too, right?
If multiple apps pull from the same web map, any changes to that map will propagate to all apps. You may have to open the app and tweak anything that doesn't automatically carry over, which items require manual updates depends on the app platform and the specific components of each app, best to keep write down what does and doesn't update.
If you are creating a second app referencing an initial map, then the map settings will reflect in the new item. The new app, on the other hand, will have a completely separate configuration file, and will not automatically update with changes made to the source app.
Similarly, if you save a web map to a new map item, that map will no longer show any changes made to the source map.
Can I ask what the purpose of doing this would be? If you want the copy to show changes when the source is changed, how is different from just using the original map / app directly?