Typically the process to update an application is to push up a new image and then tell docker to stop the old one, and load the new one in its place (yes, I know there are 0-downtime ways, but lets keep it simple).

If you put both processes in the same container, then you cannot simply upload a new image and restart the container. This is where it gets very complicated, error prone, and not in line with what I understand to be current best practices with docker.