$cookie does not contain a cookie or a reference to an array of cookies. It's garbage.

my $cookie = $cookie1."\nSet-Cookie: $cookie2"."\nSet-Cookie: $cookie3
+\nSet-Cookie: $cookie4";
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should be

my $cookie = [ $cookie1, $cookie2, $cookie3, $cookie4 ];
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Then we can clean up your code.

my @cookies;
push @cookies, $cgi->cookie(-name => "login",     -value => "", -expir
+es => '-1d');
push @cookies, $cgi->cookie(-name => 'password',  -value => "", -expir
+es => '-1d');
push @cookies, $cgi->cookie(-name => 'id',        -value => "", -expir
+es => '-1d');
push @cookies, $cgi->cookie(-name => 'CGISESSID', -value => "", -expir
+es => '-1d');

print $cgi->redirect(
   -cookie => \@cookies,
   -uri    => $cgi->url(),
);
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or even

my @cookies = map { $cgi->cookie(-name => $_ -value => "", -expires =>
+ '-1d') }
   qw( login password id CGISESSID );

print $cgi->redirect(
   -cookie => \@cookies,
   -uri    => $cgi->url(),
);
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