When renamed, Ender Chests' GUI shows "Ender Chest" instead of the new name.

The ender chests are not tile entities and can't be named the same way others can, and also what would happen if you named two ender chests different things?

that would mean that they would link up, and that would allow for multiple storages! :0

Same problem with anvils

Would be great for my new adventure map if you could rename them and all with the same name should share one inv. So a player could have his personal "Enderchest" and all players could use shared Enderchests when they are named.

I thought ender chests had to be tile entities in order to do the special particle effects? I'd say naming should have no effect on the contents of the chest if it ever is added. Separate enderchests should be through a separate nbt tag.

the ender chests are tile entities, their inventories just aren't stored in them.

a custom name tag could be saved in the chest's tile entity as it is for regular chests, and then displayed on the gui, changing depending from what ender chest block the player tried to access without even touching the inventory though.

in reply to Nathan from mojang : What would happen if you rename two ender chests ? 

I'd like to talk about just customization here. 
Just like the shulker boxes, i'd love to have an ender chest I called "Adventure Chest", or "Tiny Box Tim" to take with me on an adventure. To distinguish it from all the others.
Even if they were named, they still are ender chests. Linked up between each other. 

Last time I checked they do are tile entities.