Remove [[ and ]]
since that is used for evaluating an expression:
if kill -0 $(<pid_stored) 2> /dev/null
then
echo "PID IS RUNNING"
else
echo "PID IS NOT RUNNING"
fi
PS: You can avoid cat
also with $(<pid_stored)
To negate you can use:
if ! kill -0 $(<pid_stored) 2> /dev/null