The problem is we have an archetype of a person's life: a person is born then lives for a certain amount of time then dies. We are very used to thinking that way, so used to that idea that we think it must be that way. Our idea about "me" is very strongly tied to that archetype. Really though that's not the way it must be. It's a matter of circumstance that it happens to be that way. As we change those circumstances our concept of "me" must also change.
It's not just that. A specific consciousness is a real thing. It can only be housed in one location at a time. If you want *that* consciousness to be a different place at a different time, then you need to pick up that brain and move it there.
Now, we lose continuity every time we fall asleep. And if we were to have copy/cloning, a subset of people would very quickly adapt because it wouldn't be dissimilar enough from falling asleep and having someone move you. Experientially, anyway.
But, outside of falling asleep (etc.) consciousness does have continuity. If you lie someone down and copy their brain, they'll not be continuous with the clone. Their consciousness wouldn't split, there'd just be a new person.
Now, a strong case could be made that if all the quantum information was captured and sent (necessitating the destruction of the original), then the reformed consciousness is the same consciousness. But most 'copying' scenarios don't include that level of fine-tuning, and they all the multiple-copy outcome.