What is the name of this "wardrobe" I use for putting the student's material and books?
Closet? Wardrobe? Cabinet?
The general word is cabinet
A cupboard with drawers or shelves for storing or displaying articles
When used as you show, it's generally called a supply cabinet or a storage cabinet. Wikipedia has an article for stationery cabinet that describes the same thing, but I don't think I've ever heard that term (and the article is missing citations); it could be regional. I can't find much in the way of dictionary-style citations for my phrases, either, but googling them brings up lots of retail sites that sell products with these names.
If it's a small room, rather than a standalone device, it's then called a supply closet. Often a supply closet will contain several cabinets.