I don't really think you're correct on that one. To begin with, what makes us whole is mostly memories and our subconcious.

What makes it obvious are illnesses like dementia and general decay of neural activity as we get older.

If we implant the same memories into something or someone that person/entity would become you, like waking up from a long nap. You might not feel the same, think the same, or act the same, but it would still feel like you. Going one step further and growing an exact clone can probably go even further than that.