I have never heard people refer to shoes or hats as clothing. Clothing/clothes are typically shirts, shorts, pants, underwear, socks, dresses and skirts, etc.

This is not a stupid question at all! I understand why you would be confused about this in particular. The root word in these is "cloth". Shoes are not referred to as clothing or a piece of clothing. They're not made of cloth, so they're just referred to as shoes. "Clothing" and "a piece of clothing" both talk about shirts, pants, undergarments, dresses, and the sort. I don't recall hearing hats being called "a piece of clothing", so it may sound odd to call it that. Hope this helped a bit!

You can see a useful note on the OALD (Oxford Advanced Learner's Dictionary) that I'll paste here: Clothes or clothing? Clothing is more formal than clothes and is used especially to mean ‘a particular type of clothes’. There is no singular form of clothes or clothing: a piece/an item/an article of clothing is used to talk about one thing that you wear such as a dress or shirt.