A storage system in one embodiment comprises at least one storage array comprising a plurality of storage devices. The storage array is configured to communicate over one or more networks with a plurality of host devices. The storage array is further configured to receive one or more commands from a given one of the host devices directing that a particular user-defined name be assigned to a particular one of the storage devices, and responsive to receipt of the one or more commands, to determine if any other user-defined name is already assigned to the particular storage device, and if there is no other user-defined name already assigned to the particular storage device, to assign the particular user-defined name to the particular storage device and to store information characterizing the assignment of the particular user-defined name to the particular storage device in persistent memory of the storage array.