Azure Tips and Tricks - How to move Azure Storage Blobs between containers

In the context of moving data within a single storage account, I understand that this operation is a server-side copy, but it is still wasteful (time and IOPS cost) to duplicate all of the content and then error-prone to require a manual deletion at the end.

What is the best way to restructure content (within a single storage account) without duplicating it all and then deleting the source?

Is this also applicable to moving files between folders with Azure Files, or only blob storage (assuming we want to use azcopy or storage explorer - not SMB)?