I have not been able to rename items in an anvil on my realm server. This has happened on three occasions. They were a name tag, a grass and a pickaxe.

For the name tag and grass, naming was the only intended purpose, but I also tried to apply an enchantment book (mending) to my pickaxe and the enchantment applied, but the name did not.

Reopening Minecraft has worked to solve the problem, but it should be mentioned.