We can describe uploading as taking 3 steps.
Step 1 is reading the state of a human mind.
Step 2 is storing the state of a human mind.
Step 3 is running the state of a human mind.
The easiest way to make uploading be a move and not a copy is to make reading destructive. In order to read the state of a human mind, you have to kill the brain it is running on.
You put it to sleep, fill it full of chemicals that stop its continued processing, drop it down to near absolute zero. Then you scan it micrometer by micrometer, capturing both the layout of neurons, the state of neurtransmitters, the internal structure of neurons. You do this at a certain depth - then you literally physically shave off half of that depth, and repeat.
This process literally shreds the physical brain.
The result is then processed. The parts that matter are abstracted (at the individual neuron level) and the parts that don't matter are discarded to produce a model of the person's brain. This is then run in computer hardware.
Once uploaded, the state of a human brain might be easy to duplicate. But we may be unable to print a new biological brain -- so you cannot be both an upload and an biological brain.
We may even be able to make brain prosthetics; brain-shaped devices that store an upload and let them interact with a biological body. The path to full upload may have involved building partial brain prosthetics for brain damaged people or to deal with neurological degenerative problems.
Inventing a sciency reason why you cannot duplicate human minds once uploaded is extremely difficult. "Quantum" handwaving, where the human brains state is somehow this fragile Quantum thing, isn't very plausible; our brain is warm and wet and durable, all of which make storing an "important" distributed fragile Quantum state implausible.
A duplicate of the mind state would share a common memory and behavior.
But I can try.
It is possible that the state of a human mind is far greater than what we can store using conventional computation. At the technology level they have, storing a single human mind using conventional computation requires a trillion dollar supercomputer.
However, storing the human mind using an exotic computation system is far cheaper. But that exotic storage is fragile, far more fragile than a human brain.
So we do the shave-thing to upload the human mind, but we can't do simplifying abstractions on it: we are forced to store insane amounts of information at the individual neuron level to generate a functioning copy of the consciousness that existed before. What more, we need to simulate things at that individual neuron level to an insane degree for a conventional computer.
The actual hardware we run uploaded minds on instead stores a kind of hologram of the brain. It refactors the entire brain's storage using a (here is that word) quantum computer, so each part of the mind host is running multiple parts of the brain it uploaded.
This process compresses the mind, reducing the storage required by a factor of $10^{12}$, from world-GDP supercomputer to wrist watch. Similarly, it reduces the computation required from world-GDP supercomputer at sub-realtime (a conventional simulation) to being able to run it at slightly faster than realtime (further speedups are possible, but require exponentially more expensive hardware -- like, every 100x increase in hardware cost gives you 5% faster runspeed).
These quantum-compressed minds cannot be copied. The process of producing a copy is destructive of their state, and unlike a biological mind you can't safely put them to sleep and freeze them.
However, at the point of upload, multiple copies can be made. But once you have done that step, each copy has to run (some can run at fractions of realtime, others slightly faster) or they will decay.
And if the machine running your uploaded consciousness fails, you die.