The option to save a layer is available for a CSV added from a file but not for a CSV added from the web.
Last Published: October 12, 2015ArcGIS Online
Bug ID Number
September 9, 2015
August 21, 2020
Operating System Version
After review by the development team, it has been determined that this issue is related to a known limitation with the software that lies outside of Esri's control. The issue's Additional Information section may contain further explanation.