Applying each regex in one array to each string in another array:

my @s; # string array
my @r; # regex array
my %m; # matches hash

%m = map {
    my $k = $_;
    $_, {
        map {
            my $s =()= $k=~/$_/g;
            $_,$s
        } @r
    };
} @s;
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The =()= thingy? It forces list context on the right hand side and returns the number of elements of that list for a scalar on the left hand side, if the left hand side is a scalar. See also Re: Hidden features of Perl (More Secret Operator References)

For

counts the number of times that each string in one array matches a regex containing that string in another array

which I read as matching the regex against the string, just swap the arrays.