"Add existing module" should have created a copy of the module with all the same settings and content. It is in fact the same instance of the module merely displayed on another page. Once copied liek this, the content will always be the same in both places as will certain settings such as permissions. However, items in the Page Settings section of the Module Settings can be changed per page e.g. container andicon options. Try doing the same thing with another module on your site to confirm that injecting modules across pages is in fact working. Then check with the developer if it isn't behaving.

If you use the copy from page option, there are tick boxes to choose the behaviour and if you choose Reference, it is supposed to be like Add Existing Module" above.

Creating a new page via a page template will only copy content in modules that support it. Many third party modules do not.

Perhaps cross check on another portal if it's not working with any module.

Rob