Yes, as it's a static fact, using 'was born' is all that's necessary [passive voice] than using the past perfect in this case...

This is reported speech, therefore, you must backshift the verb before the original direct speech. So, since we never say, "I am born..." we can assume that the direct speech that he said is "I was born..." Therefore, when we report what he said, we must backshift the simple past to the past perfect: "He said that he had been born..." https://www.grammarbank.com/reported-speech-tense-change.html

They are both grammatically correct, but "he was born in 1997" is a more common useage

He said that he was born in 1997 sounds more natural