Looks like switching from
my $regex = '(2[0-4]|1?[0-9])?[0-9]|25[0-5]';
to
my $regex = qr/(2[0-4]|1?[0-9])?[0-9]|25[0-5]/;
seems to fix it. I don't immediately see why though.

It's a regex metacharacter/operator precedence issue.

The regex  | (alternation) operator has a low (the lowest?) precedence among regex operators. When a raw string like
    my $regex = '(2[0-4]|1?[0-9])?[0-9]|25[0-5]';
is interpolated into
    /^$regex$/
the final regex becomes
    /^(2[0-4]|1?[0-9])?[0-9]|25[0-5]$/

The  ^ start-of-string assertion is effectively grouped and evaluated with the  (2[0-4]|1?[0-9])?[0-9] expression and disconnected by the alternation from the  25[0-5]$ expression. IOW, the regex will match any string with a  [0-9] at the minimum (everything else is optional) at the start or with a  25[0-5] at the end, and nothing else in the string matters!

c:\@Work\Perl\monks>perl -wMstrict -le
"my $regex = '(2[0-4]|1?[0-9])?[0-9]|25[0-5]';
while (<>) {
    chomp;
    if ($_ =~ /^$regex$/) {
        print qq{'$_' matched};
    } else {
        print qq{'$_' did not match};
    }
}
100
'100' matched
z100
'z100' did not match
z255
'z255' matched
z250
'z250' matched
100z
'100z' matched
99
'99' matched
9999999
'9999999' matched
99Yikes!99
'99Yikes!99' matched
1
'1' matched
11
'11' matched
111
'111' matched
22
'22' matched
222
'222' matched
33
'33' matched
333
'333' matched
[download]

In contrast, choroba used a  qr// operator to define the  $regex object (in fact, a Regexp object). (Update: See  qr// in Regexp Quote-Like Operators in perlop.) This is not the same as a raw string! Among other things, the  qr// operator adds a non-capturing  (?:pat) group around the whole expression that, in this application, effectively preserves the desired association between start- and end-of-string assertions after interpolation:
    my $regex = qr/(2[0-4]|1?[0-9])?[0-9]|25[0-5]/;
becomes
    (?:(2[0-4]|1?[0-9])?[0-9]|25[0-5])
and is interpolated into
    /^$regex$/
as
    /^(?:(2[0-4]|1?[0-9])?[0-9]|25[0-5])$/
which can be read as "start-of-string, then one of a set of alternations in the range 0-255, then end-of-string" and which gives the desired number range discrimination.

c:\@Work\Perl\monks>perl -wMstrict -le
"my $regex = qr/(2[0-4]|1?[0-9])?[0-9]|25[0-5]/;
  while (<>) {
      chomp;
      if ($_ =~ /^$regex$/) {
          print qq{'$_' matched};
      } else {
          print qq{'$_' did not match};
      }
  }
"
0
'0' matched
1
'1' matched
100
'100' matched
1000
'1000' did not match
25
'25' matched
255
'255' matched
256
'256' did not match
a1
'a1' did not match
1a
'1a' did not match
11
'11' matched
111
'111' matched
222
'222' matched
333
'333' did not match
[download]

Bottom line: Wherever possible, prefer  qr// to raw strings for regex expressions.

Please see perlre, perlretut, and perlrequick.

Update: Incidentally, the regex  qr/(2[0-4]|1?[0-9])?[0-9]|25[0-5]/ does not match the strings  000 001 012 etc. (Update: The regex does match  00 01 02 etc.) If this is an issue, I suggest
    qr{ [01]? \d? \d | 2 [0-4] \d | 25 [0-5] }xms
instead, but whatever you use, verify it with something like Test::More as choroba did!