Renting computers as a service does not significantly change the decision to use separate data disks.
Of course you can change the defaults, why else would the API exist to create and add additional disks to instances.
One disk is simpler to manage. Especially for relatively static images where the instance is the OS and application install, not much dynamic data.
Preventing file system full remains useful. Although solutions other than multiple physical disks are possible. Separate logical volumes with LVM. Or centralized logging or messaging such that some instances do not have growing data files.
Exceeding quotas on IOPS and size may require combining multiple disks into logical volumes. (At least the quota tends to be well defined in cloud, even if the physical array remains mysterious.) Scale up databases exist.
Separate data volumes allows for some block level tricks. Imagine a major OS upgrade for a database instance, but no secondary storage exists to replicate to. Prepare an upgraded instance, but with no data. During downtime, dismount and detach data volumes, present them to new instance, and mount. Fast upgrade, no data copying, no second copy of the volume.