RMAN log file with date and name in Oracle 11gR2 in Linux
If I were doing this in the bash shell on Linux, I'd use the date command: 

$ date '+%y%m%d-%H%M%S'
20191115-111213

$ LOG_PATH="/u01/backup/logs"
$ UNIQUE="NAME"

$ rman target / LOG="${LOG_PATH}/rman.${UNIQUE}.$(date '+%y%m%d-%H%M%S').log" &lt;&lt;EOF
. . . 
EOF

$ ls ${LOG_PATH}
rman.NAME.20191115-111314.log


Remember that you'll need to clean up all these log files, from time to time.