I am not really sure Patroni has anything to do with the persistence of data. It just uses etcd, zookeper or consul to elect a master in case of a failover and uses their key value store to save the information about the current master. Its upto you whether you keep your database on a container or not. And how the data is managed by the container.
This talk by Josh Berkus explains how Patroni works pretty well.
https://www.youtube.com/watch?v=OH9WSEiMsAw
The great thing today is there are lots of options with Patroni, it is very much up to you and your requirements what kind of storage you select and how you want to recover from node and storage failure.
Patroni with Spilo e.g. relies on EBS or local discs and can also ship WAL to S3. On Google it uses their equivalent solutions.
On Kubernetes earlier posts are correct that you may rely on Kubernetes bringing your pod back with the same underlying volume thus providing some kind of availability, but for others this is not good enough, e.g. remember EBS is single AZ only. Patroni can help here, running and orchestrating either slaves or giving you automated recovery from S3.
