Minecraft wiki entry for Anvil, associated text: A name tag must be renamed before it can be used.
Renaming a bucket of fish or axolotl renames the mob inside as well, meaning a fish or axolotl can be named without a name tag.‌[Java Edition only]
A renamed item (can be any item, doesn't need to be a weapon) that kills another player or tamed mob causes the name to appear in the death message.‌[Java Edition only]
A renamed spawn egg produces a mob with the same name.
Chests, trapped chests, shulker boxes, furnaces, hoppers, droppers, dispensers, minecarts with chests, minecarts with hoppers, enchantment tables, barrels, smokers, blast furnaces and brewing stands display the name in their GUI when placed.
Renamed command blocks use their name in chat messages instead of [@].