A compose file is just the configuration of one or many containers. The container is downloaded from the chosen registry and pretty much does not get touched.

A compose file 'composes' multiple containers together. Thats where the name comes from.

When you run multiple databases then those run parallel. So every database has its own processes. You can even see them on the host system by running something like top or htop. The container images themself can get deduplicated that means that container images that contain the same layer just use the already downloaded files from that layer. A layer is nothing else as multiple files bundled. For example you can choose a 'ubuntu layer' for the base of your container image and every container that you want to download using that same layer will just simply use those files on creation time. But that basically does not matter. We are talking about a few 10th or 100th of MB in extreme cases.

But important, thoses files are just shared statically and changing a file in one container does not affect the other. Every container has its own isolated filesystem.

I understand the architecture, I'm just not sure about how docker streamlines separate containers running the same process (eg, mysql).

Quite simple actually. It gives every container its own environment thats to namespacing. Every process thinks (more or less) it is running on its own machine.

There are quite simple docker implementations with just a couple of hundreds lines of code.