Entities (tested with Minecarts (MinecartRideable, MinecartChest, MinecartFurnace), specifically) can load the CustomName tag, and when broken the dropped item has the name, but when placing the item back or saving a world with named entities, the tag is not written. The CustomNameVisible tag has no effect for entities.

If the tags are only intended for mobs, this should be broadened for all entities.