Army reservist Edward Maher died alongside Lance Corporal Craig Roberts after they collapsed in soaring temperatures while scaling south Wales's highest mountain Pen Y Fan in the Brecon Beacons on July 13.
Naming Mr Maher, whose identity had not been known until now, a Ministry of Defence spokesman said: "Our thoughts are with his family and friends at this difficult time.
"Investigations are being carried out by Powys police and the army. We cannot comment any further until these investigations are complete."
In a brief statement released through the MoD, Mr Maher's family said: "At this time of great sadness, Edward's family has asked that they be left alone to grieve in private."
Emotional tributes have been paid to a third army reserve soldier who died from injuries sustained during SAS selection training.
One of two Army reservists killed in a military exercise in the Brecon Beacons on Saturday has been named as Lance Corporal Craig Roberts.