There's no need to believe in an immaterial soul to think that a copy is a different 'you'.
It is enough to understand that having an independent "command & control" structure (for lack of a better word; C&C) is by definition 'you'. C&C means that the individual can perform a self-contained sense-think-act cycle.
We can infer this from the following:
1. Identical twins, despite having identical genetic structures, have separate C&C, and therefore qualify as separate persons.
2. Two siamese twins conjoined by the body (including parts of the head) have two C&Cs, and therefore qualify as two persons (with varying level of bodily control per person).
3. An individual with additional parts of the body (e.g., in a special case of being conjoined with a twin) has only one C&C and therefore qualifies as one person.
4. Clones are a special case of identical twins, and still have separate C&Cs, therefore qualifying as two persons.
5. A person in a coma or with another condition nonetheless qualifies as a person with some sense-think-act autonomy, despite being dependent on someone else for continued bodily functions.
6. An individual with a transplant (e.g., heart) or an extension (e.g., pacemaker) is nonetheless a person because of their consistent C&C.
7. An individual with extreme memory loss remains a person with C&C (in brain, body, genetics for the most part).
Any other special but naturally occurring cases (e.g., individuals with two brains, individuals with completely separated hemispheres, conjoined twins sharing a brain) would require either that we:
a. understand how brains and bodies work (and therefore, make a more quantified measure of how much of sense-think-act autonomy qualifies as personhood); or
b. decide based on a heuristic (1 brain = 1 person) or principle (protection of bodily autonomy, i.e., 1 body = 1 person).
But none of these need you to believe in a soul, in order to think that a digital clone is not 'you'. Unless, of course, you can prove that an individual either:
1. can sense-think-act with both physical and digital 'bodies' simultaneously and/or sequentially*, which share an equivalent experience (i.e., central C&C coordinating distribution or maintaining strict consistency across bodies).
2. can sense-think-act with either physical or digital 'body' at will, upon establishing a 'connection' and successful transfer of state (i.e., remote access).
3. can transfer whatever is necessary for C&C to another 'body' (e.g., brain transplant), and demonstrate beyond reasonable doubt that their C&C signatures in new body match the ones in the old one.