The problem you face is one that is already several thousand years old, and has never been successfully answered. The particular problem you are facing is most famously posed as the Ship of Theseus:
First, suppose that the famous ship sailed by the hero Theseus in a great battle has been kept in a harbour as a museum piece. As the years go by some of the wooden parts begin to rot and are replaced by new ones. After a century or so, all of the parts have been replaced. Is the "restored" ship still the same object as the original?
Second, suppose that each of the removed pieces were stored in a warehouse, and after the century, technology develops to cure their rotting and enable them to be put back together to make a ship. Is this "reconstructed" ship the original ship? And if so, is the restored ship in the harbour still the original ship too?
These issues plague all cloning science-fiction stories because they push the boundaries of what is meant by "self." What is an identity. Another famous one is a series of questions relating to teleporters, which is closer to your
Humanity has invented a teleportation mechanism. You enter the machine, all the information about your body and mind is copied and sent via radio waves to the receiver. Let's say the receiver is on Mars. There, the receiver builds an exact duplicate of you, down to the tiniest detail. Once the copy is complete, a signal is sent back and the transmitter on Earth destroys the body on Earth. Now there is only one of you, and it is on Mars.
What happens if the "complete" signal is disrupted. Which body is "the real you?" The one on Earth or the one on Mars?
Now consider the case where there is a mixup. Your data is sent to both Mars and Venus. The body on Earth is destroyed. Which is the "real you?"
These questions don't have unassailable answers. However you answer them, I'll give you questions that make you ponder if the other one is the right answer. The concept of "self" simply gets muddy.
A real life example is the famous conjoined twins, Abby and Brittany. Are they one person or two? Officially, we treat them as two individuals, and I would be wary of the implications of declaring them to be one individual if they did not want to be treated as such. However, they show some fascinating behaviors that make one ponder. Technically Abby, as the left individual, has control over the right side of their conjoined body. However, if Brittany wants something on the left side, such as a shoe or a cookie, there is a curious tendency for the left side of the body to reach out and grab that shoe or cookie before Brittany has even asked Abby for it. Such behaviors are those we associate with "one individual." They're the behaviors we expect from a good surgical nurse, who appears to predict what the surgeon will need even before they ask for it. These are behaviors which leave us pondering whether our simple concept of "self" is good enough to describe these complicated situations.
So the reality is that this is more complicated, and there are no easy answers. In theory, the clone might focus on some "we are one" definition of self where all individuals with the same traits are treated as one. Or they might not.
There is one thing I'd poke at, which is the idea that the cloning is perfect. It must, by necessity, be imperfect. The position of the clone when it is "created" must be different than the original's position. It will be impossible to create two individuals with precisely the same physical location. So there will be differences. The question is merely whether or not those differences are enough to cause the clone to act differently than our naive sense of "self" would suggest it behaves.