I do have a question though, why does that work, since it is not !~ and is =~?

It reads more or less "it's true if the string contains (=~) something other (^) than letters and numbers (a-zA-Z0-9)", so it works.

You can of course use !~ as well, to say "it's true if the string doesn't contains (!~) one or more letters and numbers ([a-zA-Z0-9]+) in its entirety (hence the ^ and $ that constitutes the start and end marks).

print "bad string: $string\n"
  if $string !~ /^[a-zA-Z0-9]+$/;
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which I prefer to write (yes back to =~ and not !~):

print "bad string: $string\n"
  unless $string =~ /^[a-zA-Z0-9]+$/;
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