Hi,
This seems to depend on countries and how things are agreed.
In Dubai, identification was done on departure by APP and it was stated in LoA that identification is transferred from APP to ACC. Similarly with arrivals identification was transferred from ACC to APP.
In Finland, radar contact is said on each and every reply to an initial call even though every unit uses the same surveillance system. An exception to this is when the transfer is between positions within the same unit.
Transfer of identification can be found in PART-ATS, AMC3 ATS.TR.155(c)(1) ATS surveillance services