So, I tried putting a name tag into the anvil in Minecraft Java. It took it, but wouldn't let me change the name. I tried reloading the game, but still it didn't work. I even exited out of Minecraft and went in again, but still didn't work. I even created a new world and tried it there. Still wouldn't work.
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Thank you for your report!
We're actually already tracking this issue in MC-173725, so I've resolved and linked this ticket as a duplicate.
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Has happened to me as well. I am using the new snapshot (20w10a).
The same goes for anything you can rename in an anvil!