Isn't it quite normal for an "app" to consist of multiple processes? Maybe a web server, app server, database system, message queue, object store, cron jobs and the like.
Most of the time these processes only need the network to communicate with each other. When there is a legitimate reason for processes to share the same filesystem, process namespace or network address (for example for debugging), that's when `docker exec` becomes helpful.
Certainly, I think of an SOA app being composed of multiple services, and I'm pretty sure the thing you're suggesting that the thing you want one of in the container is what I'm calling a service.
But I thought the whole idea of Docker was I could download something like a "Wordpress image" from the Docker marketplace, and it would just work in a self-contained way. Unless there is some higher level tool to download multiple Docker images for different apps and configure them together to provide a service of some kind.
For now you need 3d-party tools to define their own multi-container constructs, like Fig or Kubernetes. Eventually Docker needs to implement a multi-container construct of its own, to avoid fragmentation (interoperable containers are not as useful if they're bundled in non-interoperable wrappers). Luckily, the authors of Fig work at Docker and we're working closely with the authors of Kubernetes. So if you use either of those tools, you will probably be familiar with the native Docker version :)
If you want to start multiple separate processes, then you'll need to use some sort of init system. This is not really any different than Linux itself.