To demonstrate the process, I prepared two files, a and b (it would have been kind of you to prepare them for us).

File a contains abbreviations plus their type (day or month), tab separated:

Sun    d
Mon    d
Tue    d
Wed    d
Thu    d
Fri    d
Sat    d
Jan    m
Feb    m
Mar    m
Apr    m
May    m
Jun    m
Jul    m
Aug    m
Sep    m
Oct    m
Nov    m
Dec    m
[download]

File b contains the full names:

Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
January
February
March
April
May
June
July
August
September
October
November
December
[download]

Perl makes it possible to keep several files opened at the same time, some for reading and some for writing. Let's only output the month to two output files. Note the checks that both the input files end at the same line, i.e. none of them is shorter.

#!/usr/bin/perl
use warnings;
use strict;

sub is_month {
    my ($line) = @_;
    return $line =~ /\tm$/ ? 1 : 0
}

open my $in_a, '<', 'a' or die "a: $!";
open my $in_b, '<', 'b' or die "b; $!";

open my $out_a, '>', 'a.out' or die "a.out: $!";
open my $out_b, '>', 'b.out' or die "b.out: $!";

while (my $line_a = <$in_a>) {
    my $line_b = <$in_b>;
    die "File b shorter!\n" unless defined $line_b;

    if (is_month($line_a)) {
        print {$out_a} $line_a;
        print {$out_b} $line_b;
    }
}
close $out_a;
close $out_b;

die "File a shorter!\n" unless eof $in_b;
[download]