In the latest changelog (1.6.0.5) it is mentioned that the cloning of containers will keep their custom names. That is sadly not accurate as the cloning command still erases the custom name when using dispensers/droppers/hoppers/furnaces. It'll work fine with chests, shulker boxes and their variants.
Edit: after some more testing I found that resetting the world will rename the cloned container
Comments 2
Cleaning up old tickets: This ticket has not been updated recently (~1 year+) so is being closed as Cannot Reproduce. If you feel this is still a valid issue then please comment, or create a new ticket following the Issue Guidelines.
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Fixed in 1.7.0.2