Does pid file work by assuming pid is not usable?

Question

To use a pid file for ensuring singleton running instance of a program, I thought that when a process finishes running the program, it should remove the pid file, until I know it doesn't when reading https://stackoverflow.com/a/7453411/156458:

#!/bin/bash

mkdir -p "$HOME/tmp"
PIDFILE="$HOME/tmp/myprogram.pid"

if [ -e "${PIDFILE}" ] && (ps -u $(whoami) -opid= |
                           grep -P "^\s*$(cat ${PIDFILE})$" &> /dev/null); then
  echo "Already running."
  exit 99
fi

/path/to/myprogram > $HOME/tmp/myprogram.log &

echo $! > "${PIDFILE}"
chmod 644 "${PIDFILE}"

Here's how it works: The script first checks to see if the PID file exists ("[ -e "${PIDFILE}" ]". If it does not, then it will start the program in the background, write its PID to a file ("echo $! > "${PIDFILE}""), and exit. If the PID file instead does exist, then the script will check your own processes ("ps -u $(whoami) -opid=") and see if you're running one with the same PID ("grep -P "^\s*$(cat ${PIDFILE})$""). If you're not, then it will start the program as before, overwrite the PID file with the new PID, and exit. I see no reason to modify the script

Does the above script assume that no pid is recycled?

Can it happen that a process finishes running the program, and its pid is reused by a new process running a different program? In such a case, the quoted script will falsely think that a process is running the program.

Thanks.