Uniquely identifying java processes with same name

First, why do you want to kill any of them? If a process is using too many resources you can just get the PID in top:

top -n 1

If they need to be restarted on a periodic basis or to react to outside stimuli, use process management:

while true
do
    java -jar MyApp.jar &
    my_app_pid=$!
    while ! whatever_makes_me_think_my_app_should_be_restarted
    do
        sleep 60
    done
    kill $my_app_pid
done

Changing parameters makes a difference to the application thus it's not the best idea. You could change $0 i.e. the program name itself. Most programs don't care about that but some change their behaviour based on that (e.g. if you can call the same binary by different names like IIRC with mtools and busybox):

bash -c 'exec -a mysleep sleep 100'
ps aux | grep sleep
hl       20818  0.0  0.0   5732   576 pts/10   S+   10:01   0:00 mysleep 100

or you use the environment:

bash -c 'IDSTRING=PROC1 exec sleep 100'
PID=20936
tr '\0' '\n' </proc/"$PID"/environ | grep IDSTRING
IDSTRING=PROC1

or you start the process with a wrapper script which writes the PID and additional information (for telling the processes apart) into a log file.