The meaning is the same. The past perfect is used because his birth was before the time at which he is recalling its occurrence. The question mark is correctly placed outside the quotation because you are asking the question and the quotation is not itself a question.

"He" probably refers to two people here. Let's call them A and B. Let's rewrite it like this: “It must be around that date, since A was fairly sure that B's age was thirty-nine, and A believed that B had been born in 1944 or 1945”. The reason I think that interpretation is probable is that if A and B were the same person then he would surely be certain of his own exact birthday. The meaning is the same but the statement is more precise and clear if you say "had been born". The statement asserts that according to A the interval between B's birth and "that date" is 39 years. Since "that date" is in the past, and since "that date" is central to the discussion, the birth date being 39 years before "that date" merits the use of the past perfect tense.

Changing "he believed that he had been born" to "he believed that he was born" would not significantly alter the meaning of the sentence. Both convey the same information that the protagonist thinks he was born in either 1944 or 1945. Regarding the placement of the question mark, it depends on the context of the sentence. In American English, the convention is to place the question mark inside the quotation marks, regardless of whether the punctuation is part of the original quotation or not. In British English, it is generally acceptable to place the question mark outside the quotation marks when the punctuation is not part of the original quotation. However, in this particular sentence, the question mark would be placed outside the quotation mark since the sentence itself is not a question.