The literal translation of that is an area of disagreement between many Protestants and the Catholic Church. The meaning of the term in John 3:3 does not mean "again", but means a regeneration; being "born from above". This word has a dual meaning so it is further clarified by Jesus.
"Jesus answered, “Amen, Amen, I say to you, no one can enter the kingdom of God without being born of water and Spirit. What is born of flesh is flesh and what is born of spirit is spirit. Do not be amazed that I told you, ‘You must be born from above.” - John 3:5-7
We may be using different translations of the Bible, but what you quoted is correct. And is the same meaning as being “born again (by the spirit)”, a second birth than the natural one.
Exactly. And from whence does this power of spirituaa rebirth come?
Only God.
So the real question is whether, in all of human history, God has made spiritually reborn only people of one religion - or not. If not, then particular religious differences are not as important as they claim to be, and a far deeper connection with God is all that matters.