Login to admin by frontend form?

What you have should work – although you should not use $_SESSION (as Tom already noted). Also ID, user_pass, and user_login are all top level in the user object.

Here’s what I would try:

$email = sanitize_email( $_POST['email'] );
$pass  = $_POST['password'];
$user  = get_user_by( 'email', $email );

if ( wp_check_password( $pass, $user->user_pass, $user->ID ) ) {
   wp_set_current_user( $user->ID );
   wp_set_auth_cookie( $user->ID );
   do_action( 'wp_login', $user->user_login );
   wp_safe_redirect( wp_get_referer() );
   exit();
}