When displaying an entity's score in chat, it will not display.
However, when using a player or a fake player, it works just fine.
What I expected to happen:
The chat would display "1" for the player and the entity

What actually happened:
The chat displays a "1" for the player and any fake player, but doesn't with selectors

Steps to reproduce:
Create a new flat world type "redstoneready", to make sure no mobs spawn
Spawn a Pig

/scoreboard objectives add test dummy
/scoreboard objectives setdisplay sidebar test
/scoreboard players set @e test 1

Now you should have two players in the sidebar that have a score of 1 for objective test
After this, do the following tellraws:

/tellraw @a {"score":{"name":"@p","objective":"test"}}
/tellraw @a {"score":{"name":"@e[type=Pig]","objective":"test"}}

This displays "1" for the first tellraw, but displays a blank line for the second tellraw
You can do another tellraw to test if the @e selector specifically is broken:

/tellraw @a {"score":{"name":"@e[type=Player]","objective":"test"}}

Displays "1" aswell, showing that this is actually a problem with entity's specifically.

From Marcono1234's comment:

I used MCP to decompile Minecraft and had a look in the corresponding class and I think I found the problem.

For some reason the game creates two net.minecraft.util.ChatComponentScore, one that contains the raw input and then one that contains the parsed name and then uses the second one to display the score.However for the parsed one it uses the getName() method. This means for an ArmorStand for example it will return "Armor Stand" which is of course neither a valid player name nor a valid UUID.

To fix this some code would be needed to make the game use the UUID instead of the name when an entity is selected.

Possible fix (1.8)

List var6 = PlayerSelector.func_179656_b(p_179985_0_, var5, Entity.class);

if (var6.size() != 1)
{
    throw new EntityNotFoundException();
}

Entity selectorEntity = ((Entity)var6.get(0));
if (selectorEntity instanceof EntityPlayerMP) {
	var5 = selectorEntity.getName();
}
else {
	var5 = selectorEntity.getUniqueID().toString();
}

This is the only code change needed. The wildcard parsing will work the way it currently is (if I understand it correctly), as the recipient can currenlty only be a player (for entities it fails as it would then use the name again). However for the future it might be a good idea to change this as well.