The "preservation" of timestamps isn't really a feature of Azure Files per se, but rather the copy tool (i.e. AzCopy, Robocopy, etc.) that copies the file. When a file is written to disk, the copy tool is responsible for setting timestamps and any other metadata on the file to "preserve" them as desired by the user.
For AzCopy try explicitly setting --preserve-smb-info=true as outlined here in the docs, it is set to false by default. There is a recent issue created for AzCopy, let me know if you are still seeing this issue and I can make sure to escalate to the AzCopy team. For Robocopy you need to use the /COPY and /DCOPY options to preserve the timestamps. Good discussion on this can be found on this thread.
Hope this helps! Let me know if you have further questions or need further assistance.
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