You are doing this
$update_field($field_key, $display_name, $assignment_id);
you need to do this
$update_field($field_key, $user_id, $assignment_id);
and if your field allows multiple selections then you need to do this
$update_field($field_key, array($user_id), $assignment_id);
actually, you may need to do the second one even with only allowing a single user to be selected. I don’t remember if ACF stores an array or not for user fields that only allow one selection.