"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.
What you're saying is perfectly grammatically correct, and in my opinion it's completely natural. All you've committed is ellipsis of the phrase fasten my seatbelt:
I always fasten my seat belt because my car won't let me not fasten my seatbelt.
This is different from the case with allow, because the verb to let takes as its complement a bare verb, without the infinitive marker to.
I'll let you leave. (wrong: *I'll let you to leave.)
The nurse won't let you not take your medicine. (wrong: *The nurse won't let you not to take your medicine.)
So when using ellipsis with the verb let, it's correct to leave off the infinitive marker to, and to place the negative not at the end if required.
Best Answer
According to these Google Ngrams, both American and British English use exactly the same more than the exact same. Here is the usage in American English:
http://ngrams.googlelabs.com/chart?content=the%20exact%20same%2Cexactly%20the%20same&corpus=5&smoothing=3&year_start=1800&year_end=2000
And here it is in British English:
http://ngrams.googlelabs.com/chart?content=the%20exact%20same%2Cexactly%20the%20same&corpus=6&smoothing=3&year_start=1800&year_end=2000
Despite its usage, the exact same is considered informal (but is not deemed incorrect) by this site at Washington State University:
However, there is a long discussion of the phrase which writes that:
Proponents of the phrase note, however:
Both phrases are redundant, and the exact same can be considered correct or not depending on which style guide one subscribes to. An American English grammar guide specifically mentions that the phrase is mostly standard, and a British English guide notes that there are similar phrases. So use depends on context: in formal writing, avoid it, but in anything else the exact same can be (again, it depends on who you follow) acceptable.