My quick suggestion would be to use index instead of this regex $data =~ /$id/

if (index($data,$id)+1){
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UPDATE:Ovid's and RMGir's soutions are much more scalable. My suggestion is more of use index instead of regexes.

Benchmark

Benchmark: timing 10000 iterations of index, regex...
     index:  3 wallclock secs ( 3.52 usr +  0.00 sys =  3.52 CPU) @ 28
+40.91/s (n=10000)
     regex: 92 wallclock secs (91.44 usr +  0.00 sys = 91.44 CPU) @ 10
+9.36/s (n=10000)
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Code for benchmark

sub regex {
    foreach my $data (@db) {
        foreach my $id (@id) {
            if ($data =~ /$id/) {1}
        }
    }
}

sub index {
    foreach my $data (@db) {
        foreach my $id (@id) {
            if (index($data,$id)+1) {1}
        }
    }
}
[download]

UPDATE 2: to reiterate Ovid's point - The benchmark of my nested loop solution converges with Ovid's solution with just a hundred entries in each of your data sets

grep

<hand_wave> These are not the monks you're looking for </hand_wave>