Is this possible? If so, how?
Yes, it is possible. You can extend the cookie expire date to one month and keep the previous user id.
If the user id does not match the user id of the account being logged into, we know the user has a double account.
I believe that your method of catching the duplicated account is not a good method. What if there is two accounts who share the same computer (which is not uncommon). Your method will account it as duplicated account as well.
Implementation Example:
Login Part
if (isset($_COOKIE['prev_id'])) {
if ($_COOKIE['prev_id'] != $current_id) {
// possible duplicated account
}
} else {
// expire in one month
setcookie('prev_id', $current_id, time() + 2592000);
}