Answer
The Join option of the context menu (right click) for a feature layer allows you to join other tables to the selected layer’s table. Raster tables do not appear in the dropdown list of layers available for joining because a raster table can only be a base table. It will only allow other tables to be joined to it, and cannot be joined to another table.
To achieve an effect similar to the one in question, export the raster table to a regular table and join this new table to the target layer.