Here's a bizarre way of doing it, taking a hint from my Prolog studies (thanks Ovid!):

sub match_count {
  my ($string, $pattern) = @_;
  my $n = 0;
  $string =~ /$pattern(?{ $n++ })(?!)/;
  return $n;
}
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Essentially, match successfully, count it, but then fail the match (an empty string can always match, so negating that always fails).