Microsoft describes a shared responsibility model. The division changes by service: a virtual machine requires different operational work from a managed application platform. Customer-controlled data, identities and configuration still need ownership.
Before deployment, we recommend recording who manages updates, permissions, backups, alerts and incidents. This prevents a task from being assumed to be Microsoft's responsibility when it belongs to the business or its provider. Review that allocation again when the architecture changes, because moving to a different service can also change the operational tasks.