Because your second regex (if ($day=~/\d{1}/)) clobbers $3. I'd rework the code somewhat to:

use strict;
use warnings;

my %datehash = (
    jan => 1, feb => 2, mar => 3, apr => 4, may => 5, jun => 6,
    jul => 7, aug => 8, sep => 9, oct => 10, nov => 11, dec => 12
    );

while (<DATA>) {
    chomp;
    my $data = lc $_;
    
    next unless $data =~ /(\D{3}) \s* (\d{1,2}) \s* (\d{4})/x;
    next unless exists $datehash{$1};

    my $month = $datehash{$1};

    printf "%04d%02d%02d\n", $3, $month, $2;
}
    
__DATA__
Oct 16 2004 11:09:19:943AM
Mar 3 2007 10:30:31:170PM
[download]

Prints:

20041016
20070303
[download]

Note the use of \s to match any white space and that a greedy match is used because it makes no difference in this context. Note too that the /x flag is used so that white space can be used in the regex to make the different parts more obvious.

printf is a cleaner way of generating the string. If you want the assign the result to a variable use sprintf instead.

For future reference it is worth giving a stand alone runnable sample such as this so others can reproduce your problem easily.