>and yet still not be the same.

You're making a fundimental mistake here on understanding substrates.

If I take a hard drive an copy it, is it the same hard drive? Well, no, we'd both agree they are two different hard drives, that's pretty easy to see.

But what if we are executing data off the hard drive. Initially it would operate as if it were the same, the data on the hard drive would have zero ability or knowledge that it was copied. As the execution continued in two different places in physical reality the state of the hard drives would change.

Coming back to people, you are never the same unless you can start taking snapshots at the plank scale. You are always changing at the chemical level. Cells die, new ones are created, organs spurt out chemicals in varying amounts that alter your mental state, you 'remember' memories and by doing so rewrite them, new information enters from your senses and changes the physical makeup of the structure of our brain via re-enforcement/de-enforcement. Simply put this idea of you is an ethereal moving target. Copying that doesn't change it, each one of them will still think they are the you that has lived up till this point. When looking at both, you'll see their lives diverge, but unless they learn about each other, each you will never know that's the case.