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The directory that gets mounted into the new container is whatever is in the /app directory on the host.

This is not Docker in Docker as I understand the term to mean: you are not running a second copy of Docker inside a Docker container, you are merely making access to the host's Docker socket available inside a container.

In the scenario you show, when you are inside a container with the host's Docker socket mounted and run docker run -v $(pwd):/app2 ..., first the shell inside the container expands $(pwd) to /app, and then it sends a request to the Docker daemon to launch a container with /app bind-mounted to /app2. From the Docken daemon's point of view, this is indistinguishable from running the same command on the host directly.