Should an activity template available in one organization be available to all organizations?

When an asset group is assigned to multiple organizations, we find that an activity template linked to the asset group in organization 1 is also available to the asset group in organizations 2 and 3.  This becomes apparent when an asset is created for the asset group in organization 2 or 3.  When I go to the activity workbench, I see the activity created in organization 1 is listed for the asset created in organization 2 or 3.  Is this intended functionality?  Thank you.