There's an old philosophical problem called the Ship of Theseus, or Theseus's Paradox. It talks about a ship that was preserved by the people of Athens as a cultural relic; over time, they took away the planks that decayed and replaced them with new, identical planks, until every part of the ship had been replaced. The question then rose: Was it still the Ship of Theseus?

There isn't a definite answer either way; it's still a philosophical problem, and a matter for debate. The Narsil/Anduril question is similar.

While it's hard to see on the screen, the text gives us more of a notion about what was done in the reforging:

The Sword of Elendil was forged anew by Elvish smiths, and on its blade was traced a device of seven stars set between the crescent Moon and the rayed Sun, and about them was written many runes; for Aragorn son of Arathorn was going to war upon the marches of Mordor. Very bright was that sword when it was made whole again; thelight of the sun shone redly in it, and the light of the moon shone cold, and its edge was hard and keen. And Aragorn gave it a new name and called it Andúril, Flame of the West.

Objects in the Rings mythos, especially those made by Elvish craftsmen, are often portrayed as having some kind of life or identity to them; when Narsil was broken, it was 'killed', explicitly its "light [was] extinguished". Andúril was an entirely new blade, made from the substance of Narsil, but with its own identity as the Sword Reforged.

There's an old philosophical problem called the Ship of Theseus, or Theseus's Paradox. It talks about a ship that was preserved by the people of Athens as a cultural relic; over time, they took away the planks that decayed and replaced them with new, identical planks, until every part of the ship had been replaced. The question then rose: Was it still the Ship of Theseus?

There isn't a definite answer either way; it's still a philosophical problem, and a matter for debate. The Narsil/Andúril question is similar.

While it's hard to see on the screen, the text gives us more of a notion about what was done in the reforging:

The Sword of Elendil was forged anew by Elvish smiths, and on its blade was traced a device of seven stars set between the crescent Moon and the rayed Sun, and about them was written many runes; for Aragorn son of Arathorn was going to war upon the marches of Mordor. Very bright was that sword when it was made whole again; the light of the sun shone redly in it, and the light of the moon shone cold, and its edge was hard and keen. And Aragorn gave it a new name and called it Andúril, Flame of the West.

Objects in the Rings mythos, especially those made by Elvish craftsmen, are often portrayed as having some kind of life or identity to them. When Narsil was broken, it was 'killed'; explicitly, its "light [was] extinguished". Andúril was an entirely new blade, made from the substance of Narsil, but with its own identity as the Sword Reforged.