Auto Login upon signup

Hi,
I need to auto-login my user on signup on my webiste, so i have used the ‘wp_set_auth_cookie’ on ‘user_register’ hook

add_action( 'user_register', 'custom_autologin', 10, 10 );
function custom_autologin( $user_id ) {
user = get_user_by( 'ID', $user_id );
	wp_set_current_user( $user_id, $user->user_login );
	wp_set_auth_cookie( $user_id );
	do_action( 'wp_login', $user->user_login );
}

So it’s working.
My issue is that When a user-1 register and is auto logged into the website, and if he closed the browser without logging-out and user-2 is accessed the website from the same browser and since the sut cookie remains two days ( wp_set_auth_cookie ) the user-2 is auto logged into user-1’s account. how prevent it ? any other option for auto logging user into website on registration ?
or any option to unset the auth_cookie on closing the browser ?

Thanks in advance