"Exact same" represents a grammatical practice that is particularly prevalent in American English; the use of an adjective for an adverb. In this phrase "exact" modifies "same" and is functioning as and adverb.
In the literal sense "exact same" is indeed redundant, however, words aren't quite so precisely defined as apparently your teach would have you believe. If I have a Hugo Bos blue shirt with an 18 inch collar, someone with a Hugo Bos blue shirt with a 20 inch collar might think we have the same shirt. In fact, someone with a Hugo Bos white shirt with a 20 inch collar might think we have the same.
You might even argue that if they are two shirts identical in every respect they are still not "the same" shirt. If I wore my shirt today, and again tomorrow, you might tell me "you're wearing the same shirt as yesterday", and that would be absolutely literally true.
Which is to say, "same" is used rather more loosely than "the identical object" in common language.
By modifying it with "exact" you are emphasizing that they are even more "same" than if you did not so modify.
Many words sound like they are absolute, binary, and not subject to gradation. However, I am reminded of a discussion between Sheldon Cooper and Stuart the comic book guy on the hilarious TV show "The Big Bang Theory":
Stuart: Ooh, Sheldon, I’m afraid you couldn’t be more wrong.
Sheldon: More wrong? Wrong is an absolute state and not subject to gradation.
Stuart: Of course it is. It’s a little wrong to say a tomato is a
vegetable, it’s very wrong to say it’s a suspension bridge.