Regular chests keep their name with silk/hand break. While the ender chest when broken with silk touch loses its custom name added with an anvil.
Videos of examples:
https://drive.google.com/drive/folders/1nScNBbZ0Q2O8LvGKUlgi7vi0FzJBl9b7?usp=sharing
Thank you for your report!
We're tracking this issue in MCPE-93918, so this ticket is being resolved and linked as a duplicate.
That ticket has already been resolved as working as intended, which means this is not considered a bug and won't be fixed. Please do not leave a comment on the linked ticket.
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