The is apparently not a browser issue. It is an issue with USING current version of Silverlight with Windows XP. Microsoft does not support XP and he requirements for Silverlight excludes  XP.
Of course, this is not a problem with Windows 7, 8 and 10.
Reference:
https://www.microsoft.com/silverlight/what-is-silverlight/

But you can get fair results with adobe flash in some browsers.

Chrome is OK when I watch a movie on my old XP laptop.
But with Silverlight I get an error message on Amazon prime.

How about you?

Ran into this issue a while back with my older XP system and trying to watch Netflix which required Silverlight. Silverlight 5 works with XP SP3. So if you can get Silverlight 5 you might be ok until the service checks the version and complains that it no longer supports 5 etc. http://superuser.com/questions/656414/is-silverlight-5-compatible-with-windows-xp-sp2Thanks. I fol;led the link and then was led  to another sight. One must pick  some new version of Silverlight, but not too new.

EDIT: Good. It now works OIK on Amazon prime.  The version that works  for me is:
Silverlight_5.1.20125.0.exe
From:
http://www.oldapps.com/silverlight.php?old_silverlight=11966?download
Nice that somebody is helping the XP crowd. 
Quote

Nice that somebody is helping the XP crowd.