That may be true for some UNIXes, but not for all UNIXes. For example on FreeBSD, create a process that will stick around for a bit:

  # cat >sleepy.c
  #include <unistd.h>
  int main(void) {
    sleep(999);
    return 0;
  }
  # cc -o sleepy sleepy.c
  # ./sleepy &

Now try to overwrite it:

  # cp /bin/date sleepy
  cp: sleepy: Text file busy

Now try with install:

  # install /bin/date sleepy
  # ./sleepy

Install was able to get around the "Text file busy" error. Why?

  # touch a
  # stat -f %i a
  686974
  # cp /etc/motd a
  # stat -f %i a
  686974

cp here has preserved the inode, it replaced the contents without deleting and recreating the file. Now lets try install:

  # stat -f %i a
  686974
  # install /etc/motd a
  # stat -f %i a
  687076

The inode changed. Why?

  46785 install  CALL  unlink(0x7fffffffed52)
  46785 install  NAMI  "a"
  46785 install  RET   unlink 0

install unlinks the file first before copying the data over. Then it creates a new file/inode (F_CREAT):

  46785 install  CALL  openat(AT_FDCWD,0x7fffffffed52,0x602<O_RDWR|O_CREAT|O_TRUNC>,0600<S_IRUSR|S_IWUSR>)
  46785 install  NAMI  "a"
  46785 install  RET   openat 4