No, it is not possible to obtain a short share URL for a web map in the new ArcGIS Online Map Viewer. To generate a short share URL for a web map, open the web map in Map Viewer Classic. Refer to ArcGIS Online: Link to a map for steps and browse through the page for more information.
In the Share window, copy the short URL in the Link to this map section.
Note: As an alternative, in the new Map Viewer, copy the long URL that displays in the web browser address bar.