I agree here: aside from meaning identical, the word "same" also means "similar", which is probably, or rather, most likely the reason speakers add the adjective "exact" here:

Ex: She's wearing the exact same outfit.

It's perrfectly grammatical. However, it is somewhat redundant, at least to speakers who read, exact = identical. Other than that, there isn't a problem at all.

About its function. Notice the position of the word "exact". It comes after the definite determiner "the", which makes "exact" nominal, not verbal. Moreover, add -ly (make it into an adverb) and the result is ungrammatical:

Ex: *She's wearing the exactly same outfit.

"exact" isn't an adverb. It's an adjective (meaning, 'strictly accurate or correct, precise;e.g., an exact likeness; an exact description. Source: Dictionary.com) which, by the way, is the reason 'Merriam-Webster's Collegiate Dictionary (11th edition) doesn't list "exact" as an adverb.' ;-)

In short, the problem here isn't "exact"; it's this assumption:

' "Same" is clearly an adjective, and "exact" modifies "same", so you would expect it to be an adverb.'

Why expect that? That's what has me stumped.

All the best. :-D