You seem to be talking about a vastly different use case.
Containerized apps having their own database? What? Aren’t these types of containers stateless? I always very much try to keep state out of app containers.
What kind of data storage are we talking about?
Those database containers get a PVC/volume/mount for their data dirs. The only thing ever connecting to them is their "owner" application container. So at that point, why not drop the postgres container and PVC mount a sqlite directory in the app container? The result is the same.
Especially since in production you might want to scale the parts separately. I like to have a Postgres cluster to connect where backup is already handled, and the app then doesn’t have any persistent data, doesn’t need any network volume mounts.