I think it didn't unlock because my internet has been terrible lately. I tried this again the next day and it worked just fine! YAY!
When naming a shulker box in an anvil the Organizational Wizard achievement doesn't unlock.
I tried doing this on two different worlds. And yes, they are capable of earning achievements. And yes I was signed into xbox live.
And in my testing I realized anvils don't make sounds when naming/repairing an item. And they reset to a fully repaired anvil when placing down a broken anvil. So this bug may have to do with anvils not functioning probably.