When you verify the student discount, you should receive an email notification with a button to complete the purchase. Unless you complete the purchase fromt he email link, the discount won't be activated.
If you've completed these steps but you were still charged for the regular Premium subscription, I suggest checking if you have other Spotify accounts following these steps. You could've created a new account or applied the discount to the wrong one.
Let me know what you find.
“Music acts like a magic key, to which the most tightly closed heart opens.” ― Maria Augusta von Trapp