Since the Pi-hole Docker image should not be upgraded the same way as a physical instance (using pihole -up), the container for each new release has to be rebuild. It's surprisingly hard to find up-to-date information on how to update the image when running Pi-hole on top of an OpenMediaVault instance. Most available information is already a couple of years old, and often refers to using additional plugins (Watchtower) or still applies to the outdated Portainer variant.
It's actually pretty simple: First, navigate to "Services"-"Compose"-"Files" in the OMV web UI. Then, selecting the "pull" button gets the latest container version from the registry. This can take a couple of minutes. Afterwards, the container needs to be taken down using the "down" button and started again using "up".
Assuming the container has been configured with volumes to store the user data, no configuration data is lost - it's an inplace upgrade just like using pihole-up.