In such scenario, verify if the Spotfire Server URL that was used to access the analysis file and URL in the new tab are same or not.
If not, specify the same Spotfire Server URL in the Public Address URL as seen in the below screenshot:
Make sure the URL specified in the above screenshot is same as you use when accessing the Spotfire Server.
Refer below knowledge base article to change the Public Address URL:
How to update the Web Player URL used while generating absolute URLs when saving analysis files as a library item with the Analyst Client. For example: If you are accessing the URL http://10.97.111.127:10000 from a browser to access the Spotfire Server, make sure the URL in the above screenshot is also the same(http://10.97.111.127:10000). Restart the Spotfire Server and the Node Manager after saving this configuration change.
After this change has been made, you will find the same Spotfire Server URL on accessing the 'View Library' option as well and you will not be prompted for login again in the same browser session.