In this case the advantage are operators for running postgres.

With Docker Compose, the abstraction level you're dealing with is containers, which means in this case you're saying "run the postgres image and mount the given config and the given data directory". When running the service, you need to know how to operate the software within the container.

Kubernetes at its heart is an extensible API Server, which allows so called "operators" to create custom resources and react to them. In the given case, this means that a postgres operator defines for example a PostgresDatabaseCluster resource, and then contains control loops to turn these resources into actual running containers. That way, you don't necessarily need to know how postgres is configured and that it requires a data directory mount. Instead, you create a resource that says "give me a postgres 15 database with two instances for HA fail-over", and the operator then goes to work and manages the underlying containers and volumes.

Essentially operators in kubernetes allow you to manage these services at a much higher level.