Is placing the deceased in the morgue until their family is identified, knowing that the soul returns to them to live in the life of Barzakh, considered torture for them, and is it permissible Islamically?

There is no harm in delaying burial for a legitimate Shariah interest, or due to the unavailability of someone to perform the burial, or to complete an investigation into the cause of death. There is also no harm in placing the deceased in a refrigerator to preserve the body from decomposition.