Multiple authentication prompts seem to be a general problem for WMP even though a user is logged in to SharePoint portal.
Why it asks for credentials? This YouTube video explains such a behavior the best way.
For workaround try the following steps:
1. Make sure the site is added to trusted sites in IE options
2. In IE options in the user authentication section set an option "Automatic logon with current username and password" for the corresponding Internet Zone.
3. If the above hasn't helped, disable the proxy settings for HTTP in WMP: Tools > Options > Network > Protocols > HTTP > set to "None"
4. Open Control Panel -> Credentials Manager on your machine and add a new Windows Credential for your site and respective user credentials.
Restart the browser.