/.../, m/.../, m#...#, m!...! and so on are all equivalent. The character following the m will be used to delimit the regexp.

Use
($reversed_name) = $string =~ m/\/(.*?)\|/;
if you prefer. It means the same thing, but it's not quite as readable.

foreach $string (
  "s005219/Doe John|John.Doe\\",
  "s0052194/Doe Thomas|Thomas.Doe\\",
) {
   ($reversed_name) = $string =~ m!/(.*?)\|!;
   print($reversed_name, "\n");
}

# Prints
# ======
# Doe John
# Doe Thomas
[download]

Alternatively,

foreach $string (
  "s005219/Doe John|John.Doe\\",
  "s0052194/Doe Thomas|Thomas.Doe\\",
) {
   if (m!/(.*?)\|!) {
      print("$1\n");
   }
}
[download]