If you add these lines at the start, you will find lots of useful messages:

use warnings; # needs 5.6.0 or newer,
# $^W = 1;    # for older
use strict;
[download]

The $search_for variable is only ever set to "Jan" because you assign it that way in my $search_for = $_[0];. I think you want to do something like this:

sub search_database {
    my $database = shift;
    my @search_for = map {qr/$_/i} @_;
    my @results;
    open(DB, $database) or die 'Error opening file: ', $!;
    while (<DB>) {
        my @record = split /::/;
        push @results, [@record]
            if grep {
                     $record[5]   # or whichever
                       =~ /$_/i;  # /o taken care of by qr
            } @search_for;
    }
    close DB or die $!;
    \@results;
}
[download]

That removes the $search_field conditions which are badly set up and are confusing the issue. The search now is restricted to just the field where a date should appear, avoiding confusion with "Janice", "Martin", and "taproot". The function now returns a reference to the @results array instead of writing to one in a larger scope. The arguments are changed to include the name of the $database file. The caller gets all fields, and can index the desired ones.

After Compline,
Zaxo