It seems the name NBT tag gets removed when a Mooshroom is converted into a regular cow by shearing. (This is probably because a new entity is spawned when the mooshroom is sheared, but I think the NBT data should remain.)

Steps to Reproduce

• Name a Mooshrrom spawn egg using an Anvil
• Spawn Mooshroom (notice name above its head)
• Use shears on Mooshroom
• Name is not shown on the cow (sheared mooshroom)