It sounds like you might have an organization particularly in mind but I can’t quite tell.
A kiruv organization that I know goes after Jews thats it. Sometimes people, like you described who think they’re Jewish, will show up. Usually the Rabbi and Rebbitzen try to sort them out. Sometimes its much later down the line. The goal is not to convert them. However, after learning and living life as a frum Yid, then the go through the process. (Which I have heard is easier/shorter if a father is halachicaly Jewish but the mother is not)