That's right. Why is that a problem?
Imagine you had this done to you, and imagine that in order to clone your brain you have to be put under anesthesia (because the process takes time and you need to capture a coherent snapshot). When you wake up, how are you going to tell whether you are the clone or the original?
As far as I’m concerned we might very well be just processes that can in theory be copied and recreated. In that case the person in the thought experiment would exist twice at a single point in time and then diverge into to different persons due to different environments and probabilistic processes in the body.
This already happens, though before the formation of any memories. We call them identical twins.
The divergence is more limited than you might imagine. One striking experiment found that if you separate identical twins and give them the instructions "just draw whatever comes into your head", they are likely to draw the same things.
that being:
> If we remove our "selves" from our bodies, are we still us?
But answering this is now a straightforward thought experiment. You just approach it incrementally.
If you replace your limbs with prosthetics, are you still you?
If you get a heart-lung transplant, are you still you?
If you get an artificial heart, are you still you?
etc.
> a somehow unique, identifiable self which ... is how most people view themselves
Well, yeah, but that's a limit of our present technology, not necessarily a fundamental aspect of what it means to be human. Technology redefines the answer to that question all the time.