The town of Bethlehem where Jesus is said to have been born is now bigger, and where the manger once was on the outskirts of the town is now engulfed within it, and is now its town's major tourist attraction. Regardless, the original manger where Jesus was born is lost to history, but are there any relics that are thought to have once been part of it, such as pieces of wood or straw? Are there any remnants of the gold, frankincense, and myrrh that were given to Jesus according to the Bible? Is there any remnant left of this historic religious site, important to both of the world's largest religions?
While you wait for a better and more complete answer, I would point out that your question is based on a flawed premise. Namely, that the birth of Jesus (as narrated in the gospel of Matthew and then in the gospel of Luke) is a historical event. Most scholars (except some apologists) agree that it was most definitely not a historical event, and indeed both the narratives of Matthew and Luke were crafted in order to remove the inconvenient, embarrassing truth of Jesus being born in Nazareth. And then Matthew also tries to fulfill as many 'prophecies' as possible and creates parallels with Moses' story.
In addition, the details portrayed in the two narratives are contradictory and not reconciliable (in Matthew's story Jesus is born in Bethlehem, his hometown, in his very house, not in a manger in a town his family went to just to do a census). The so-called 'Grotto of the Nativity' is a place chosen because of tradition, and there is no proof whatsoever that it has been Jesus' birthplace.
u/KiwiHellenist's wonderful article about the birth narratives shows many of the problems encountered in the two birth narratives.