Lightroom 6 map module "Set all to the same GPS coordinates" option always disabled/grayed out

I can't reproduce this on Mac. Can you please confirm which flavor of Windows you're using?

The "Set all to the same GPS coordinates" is enabled only when you do contextual mouse click on the icon for a "Cluster of nearby photos", which represents a group of photos at different locations (depending on the map zoom level).

To build on Simon's reply, there are three ways to set a group of photos nearby on the map to have the exact same GPS coordinates:



1. As Simon suggested, zoom out until the photos all appear as a single cluster, then right-click the cluster and select Set All To The Same GPS Coordinates:



2. Zooming out may accidentally include extraneous photos; if so, click and then shift-click each of the desired photos' bubbles on the map so that they are all orange. Then point the mouse cursor where you want all the photos, right-click and select Add GPS Coordinates To Selected Photos. Make sure you're not pointing over an orange bubble:



3. In Library, you can use Metadata > Copy Metadata and Metadata > Paste Metadata to copy the coordinates from one photo to the others.  However, there is a long-standing bug that rounds the pasted coordinates to the nearest second, which on average will be about 17 m / 15' away from the original coordinates.  So the GPS coordinates of the source and the destination photos will be slightly different.

See this bug report: 
https://feedback.photoshop.com/photoshop_family/topics/copy_metadata_loses_gps_precision?topic-reply...