Does it make sense when two articles are used, for example “An exactly the same”? If the former conveys inspecific/abstract item, then what does the latter do?
You're asking if we use the phrase: An exactly the same?
Not normally. It's not generally grammatical to use an article to introduce an adverb (exactly), nor to use two articles to introduce a noun (same).
But it could be used for some kind of special effect, as could almost any string of words. A context might be that two people see a woman. One says she's the same woman they saw on the TV news because she robbed a bank:
A: I'm telling you, she's a completely different woman!
B: And I'm telling you: she's an exactly-the-same woman. Let's catch her!
That's something I can cook up in my imagination as possible. But whether or not it's grammatical depends on a careful definition of grammar.