The short answer is Yes.
It is also used when a pilot is requested to contact the ops department on "box 2" (the 2nd radio on the aircraft, i.e "speedbird 2959, can you call company on box 2."
The "follow / after the landing / departing company" is as you say a way of controllers not having to say British Airways, British Midland, etc etc. reduces the RT workload.
As a cautionary note, it should only be used when the a/c is positively identified as another "company" aircraft and not one leased from another operator as often happens when aircraft have technical problems