To answer your question directly, I do not think it is possible to deploy a service that has two different containers: DB server container, and Web server container. This does not include scaling (service is automatically scaled to a certain number of container instances).
However, you can deploy a container (a service) that contains multiple processes, although it might not be considered as best practices, as mentioned in this article.
Cloud Run takes a user's container and executes it on Google infrastructure, and handles the instantiation of instances (scaling) of that container, seamlessly based on parameters specified by the user.
To deploy to Cloud Run, you need to provide a container image. As the documentation points out:
A container image is a packaging format that includes your code, its packages, any needed binary dependencies, the operating system to use, and anything else needed to run your service.
In response to incoming requests, a service is automatically scaled to a certain number of container instances, each of which runs the deployed container image. Services are the main resources of Cloud Run.
Each service has a unique and permanent URL that will not change over time as you deploy new revisions to it. You can refer to the documentation for more details about the container runtime contract.
As a result of the above, Cloud Run is primarily designed to run web applications. If you are after a microservice architecture, which consists of different servers running each in unique containers, you will need to deploy multiple services. I understand that you want to use Cloud Run as database server, but perhaps you may be interested in Google's database solutions, like Cloud SQL, Datastore, BigTable or Spanner.