my @nums = ('1203', '1204', '1207');
my $regex = '\b(?:' + join('|', @nums) + ')\b';
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No way is provided for  if (my ($match) = m/$regex/) { ... } to capture anything (and thus be true). Also, only a single match per line is assumed. (Also  + (addition) is used where  . (concatenation) is intended.)

I would suggest something along the lines of

c:\@Work\Perl\monks>perl -wMstrict -MData::Dump -le
"my @nums = ('1203', '1204', '1207');
 my $regex = '\b(?:' . join('|', @nums) . ')\b';
 print qq{'$regex'};
 ;;
 my @matches = 'w 1207 x 1203 y 9999 z' =~ m{ $regex }xmsg;
 dd \@matches;
"
'\b(?:1203|1204|1207)\b'
[1207, 1203]
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while ( <$file_h> ) {
  if (my @matches = m/$regex/g) {
    ++$matched{$_} for @matches;
    print "$file $_";
  }
}
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Update: Changed code example to include 9999 group.

Update 2: Shorter, IMHO sweeter:

c:\@Work\Perl\monks>perl -wMstrict -MData::Dump -le
"my @nums = qw(1203 1204 1207 1111);
 my ($regex) = map qr{ \b (?: $_) \b }xms, join ' | ', @nums;
 print $regex;
 ;;
 my %seen;
 for ('w 1207 x 1203 y 9999 z',
      'w 11203 x 12033 y 112033 z',
      'w 1207 x 1203 y 9999 z 1207 zz',
     ) {
   print qq{>$_<} if map ++$seen{$_}, m{ $regex }xmsg;
   }
 dd \%seen;
 ;;
 my $not_seen = join ' ', grep !$seen{$_}, @nums;
 print 'num(s) not seen: ', $not_seen || '(none)';
"
(?msx-i: \b (?: 1203 | 1204 | 1207 | 1111) \b )
>w 1207 x 1203 y 9999 z<
>w 1207 x 1203 y 9999 z 1207 zz<
{ 1203 => 2, 1207 => 3 }

num(s) not seen: 1204 1111
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