It is cool and useful, (like copying/pasting Syle values), but keep in mind copy/pasting the JSON for the coordinate system is basically an "assign coordinate system" or "repair coordinate system". It is not a "change coordinate system."
Suppose Drawing A, created from Table A, has the correct coordinate system assigned, some version of Orthographic. Suppose Drawing B, created from Table B, is supposed to be using that exact same version of Orthographic, but instead it somehow got Latitude/Longitude or Pseudo-Mercator assigned, so now it doesn't show up where it should.
In the drawing's table, the FieldCoordSystem.Geom property gives the coordinate system to be used for the Geom field, if that's the name of your geometry field. So, if you want to have Drawing B use the same coordinate system as drawing A, just pop open Table A's properties, copy the value of the FieldCoordSystem.Geom property, and then pop open Table B's properties and paste that into the value of FieldCoordSystem.Geom property in Table B.
Now, suppose Drawing B uses Lat/Lon and it has had Lat/Lon correctly assigned as the coordinate system to be used. Drawing B shows up in the right place, etc. You want to re-project Drawing B from Lat/Lon into the same Orthographic used by Drawing A. To do that you have to use the Change Coordinate System dialog.
If you simply copy the JSON string out of the FieldCoordSystem.Geom property in Table A, and then paste it into the FieldCoordSystem.Geom property in Table B, all that will happen will be that Drawing B will now have assigned the wrong coordinate system, since the actual numbers within the Geom itself aren't changed by that. Doing that is equivalent to launching Repair Coordinate System and saying "whoops... this isn't supposed to be using Lat/Lon, it's really in Orthographic...".