I think for anything other than what you are saying to be true there would have to be some measurable essence, making a person a person, a soul, that could only exist in one body at a time, and in and of itself be unable to be copied no matter advancements in technology. Meaning that a clone could be made, but it would not have life unless the essence allowing it to run was taken from the original which would cause the original to die.
Essentially, magic.
That is the only way I think, you could say a clone is not a clone.
But what you're really doing is kicking the can down the road and saying the soul is the person not the body they inhabit.

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"