Your pattern, /(\d)/ just means "match a single digit character," which will match the first digit in any number, regardless of length. If you wanted to do this with a regex, you could say,

if ($month =~ /^\d$/) { 
  $month = '0' . $month;
}
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Note that there is no need to capture the match with parentheses. This is a bit overkill for simply formating a number, though. Why not just use sprintf?

$month = sprintf("%02d", $month);
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Update: Corrected typo.