This is an philosophical question more than a scientific one.A copy is not the original in a literal sense. Even if there is no measurable difference between an original and a copy after the event that they split-off, the fact remains they do not share the same history. Philosophically that is a distinction that could have weight.

If humans agree that a perfect copy is functionally no different from the original despite the fact that it is a copy with a different history, then for human purposes they could be considered the same thing.And if we agree to draw a distinction that says they are different, then they are, for our purposes different.

Persistence of a consciousness can be a huge factor. If the original ceases to exist at the same time the replica is made conscious, it is very easy for a person to rationalize that there was a transference of consciousness, not the end of one and the beginning of another.

The game Soma explores this a lot. (super recommend)

The end of The Prestige explores this a bit.

Star Trek TNG explores this in "Second Chances"