my %hash = (
    foo  => ['this', 'that'],   # Two different values
    bar  => ['those', 'those'], # Two identical values
    baz  => ['the', 'other'],   # Not the same
    buzz => ['same', 'same'],   # The same
);

##</code><code>##

my %hash = (
    foo  => ['this', 'that'],   # Two different values
    bar  => ['those', 'those'], # Two identical values
    baz  => ['the', 'other'],   # Not the same
    buzz => ['same', 'same'],   # The same
);
while (my ($key, $aref) = each %hash) {
    if ($aref->[0] eq $aref->[1]) {
        print "$key\n";
    }
}

##</code><code>##

if (2==@$aref && $aref->[0] eq $aref->[1]) ...

##</code><code>##

while (my ($key, $aref) = each %hash) {
    my %seen;
    $seen{$_}++ for @$aref;
    next if 1 != keys %seen;
    print "$key\n";
}