Renamed Tile Entities don't store their NBT tag when placed.
What I was expecting to happen was:
The Tile Entity stored the information like this:
What actually happened was:
The Renamed Tile Entity stores the NBT data only when it's an item entity in inventory.
Steps to Reproduce:
1-Place an Anvil.
2-Rename a tile entity to something else.
4-Place the block
3-Browse the NBT data of the tile entity.
4-Note that there isn't any NBT data about the CustomName tag.