ArcGIS Experience Builder with the embedded ArcGIS Dashboards prompts to log in to access ArcGIS Dashboards when the user is logged in and the dashboard is not shared publicly.
Based on the description, it does look like a similar issue to BUG-000139966. However, it cannot be reproduced on any of the following ArcGIS Enterprise versions:
ArcGIS Enterprise version 10.8.1 and ArcGIS Enterprise version 10.9
There is no URL to copy from the bottom right of the Portal item page at all. Described case N/A.
ArcGIS Enterprise version 10.9.1
There is a URL on the bottom right of the item page, but there is a bug. The URL is broken and is not working properly. Described case N/A.
For all cases, the URL directly copied from the ArcGIS Dashboard app runtime works fine in the ArcGIS Experience Builder Embed widget, with no login window.