That's not what dind is. Rather, there is a docker daemon running inside the container, and the containers it hosts are nested inside its cgroup in the host kernel. The result is very close in feel to docker in its own VM.
Furthermore nesting can be done inside one of the payload containers creating a turduckin. E.g. you can run k8s in a container, with k8s nodes implemented as nested containers and the cluster pods as doubly nested "pigeon" containers.
I haven't tried more than three levels but in theory more should work.
docker-in-docker doesn't run a docker daemon in the container, it just bind-mounts the host's docker socket inside the container, and the docker client talks to that. Any containers you launch from within docker-in-docker are siblings, not nested.
What is Docker in Docker?
Although running Docker inside Docker is generally not recommended, there are some legitimate use cases, such as development of Docker itself.
...If you are still convinced that you need Docker-in-Docker and not just access to a container's host Docker server, then read on.
This makes it pretty clear that it's a different copy of the docker daemon (which eg. allows you to test changes to docker itself) and specifically says it's different from "just access to a container's host Docker server".
I haven't tried more than three levels but in theory more should work.