For each hash element, you execute

if ($value =~ /$search/gi) {
    $contains++;
}
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$contains++ isn't in any kind of loop, so it gets executed at most once. So let's add the loop we need:

while ($value =~ /$search/gi) {
    $contains++;
}
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This is equivalent, yet simpler and faster:

++contains while $value =~ /$search/gi;
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And this is faster yet (assuming $search contains no captures):

$contains += () = /$search/gi;
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By the way, using m//g in scalar context outside of while condition is a bug.* This includes using it as an if condition.

* — Yes, this is a generalisation, but it's accurate for those don't know what makes the statement incomplete.