Student discount and Teacher discount?

Nope, you cannot stack discounts.

My understanding is that the discount that Apple offers direct to customers is an Educational Discount as in they offer discounts for both students and teachers. It is the same discount either way.

Additionally many educational institutions have agreements with Apple where they are able to extend their discount to their employees and students which is different than the above discount.

If you go to a $9 movie, and the price is $6 for students and senior citizens, would you expect to get in for $3 if you're a 68-year-old student?

There is, as posted, an education price that is lower than the standard price. You can get the educational price as a student, a teacher, or in some cases an employee of an educational institution. It makes no difference how many of these you are. It is a lower price for which you need to be eligible in at least one way. Being eligible in more than one way doesn't make the education price any lower.