Killing process started by bash script but not script itself

Doing The Right Thing

Use a real process supervision system -- your Linux distribution almost certainly includes one.

Directly monitoring the supervised process by PID

An awful, ugly, moderately buggy approach (for instance, able to kill the wrong process in the event of a PID collision) is the following:

while :; do
  ./Server.sh & server_pid=$!
  echo "$server_pid" > server.pid
  wait "$server_pid"
done

...and, to kill the process:

#!/bin/bash
#      ^^^^ - DO NOT run this with "sh scriptname"; it must be "bash scriptname".

server_pid="$(<server.pid)"; [[ $server_pid ]] || exit
# allow 5 seconds for clean shutdown -- adjust to taste
for (( i=0; i<5; i++ )); do
  if kill -0 "$server_pid"; then
    sleep 1
  else
    exit 0 # server exited gracefully, nothing else to do
  fi
done

# escalate to a SIGKILL
kill -9 "$server_pid"

Note that we're storing the PID of the server in our pidfile, and killing that directly -- thus, avoiding inadvertently targeting the supervision script.

Monitoring the supervised process and all children via lockfile

Note that this is using some Linux-specific tools -- but you do have linux on your question.

A more robust approach -- which will work across reboots even in the case of pidfile reuse -- is to use a lockfile:

while :; do
  flock -x Server.lock sh Server.sh
done

...and, on the other end:

#!/bin/bash

# kill all programs having a handle on Server.lock
fuser -k Server.lock
for ((i=0; i<5; i++)); do
  if fuser -s Server.lock; then
    sleep 1
  else
    exit 0
  fi
done
fuser -k -KILL Server.lock