Quasi-Sapient Personal Extension; it’s a type of self-learning exoself plugin, and specifically the type that behaves most like having a helpful ally sharing your brain.
So, having filled in all that background, let’s return to the original questions:
In those senses, not really, for the reasons given above.
You can have a whole lot of threads and agents, some quite chatty, in your mind - notwithstanding actually carrying another mind-state around in a separate partition - as a product of highly sophisticated cognitive engineering, but that isn’t the same thing. And besides, they’re all a part of you. That’s the point. If any of them turn into an independent identity rather than a shard-identity of yours, something has gone wrong with the plan.
(Of course, if you’re prone to running experimental versions of advanced sophotechnology, something may indeed go wrong with the plan.)
It’s possible, but possible does not mean likely. Remember, no mental routine is complete in and of itself; the self emerges from the chorus of interacting routines. You, the alter, need to be able to demonstrate that you and all your supporting routines are, as a body, sufficiently diverged from you, the primary, to constitute a (technically and legally) distinct individual and not merely a glitchy subpersonality. This will involve a lot of time having your mind-state diagrammed and analyzed in depth by iatropsychists and cognitive engineering professionals.
A big godsdamned mess, mostly, seeing as when you fork and/or reproduce you’re supposed to make the appropriate arrangements in advance, especially as your alter - if a distinct individual but not a fully competent individual - may actually be your child in law, the same as clones, pruned forks, accidentally self-aware fictionals, and the like. Hope you’re good at cooperating with yourself.
(And no, they don’t get to be an automatic citizen-shareholder. For one thing, citizen-shares represent an investment in the Empire that covers the attached Citizen’s Dividend; they can’t be split, therefore, without reducing the value of everyone else’s citizen-shares, and that’s not happening. Prospective parents by any mechanism are, by custom, supposed to escrow the cost of purchasing their childrens’ shares in advance.)
Mind-editing is a requirement, and it’s going to require a highly skilled psychedesigner to disentangle the complex processes, emphasis on the plural, of identity in a case like this.
So possible, yes. Easy, no, for values of not easy mostly equal to expensive. Picture the psychedesign equivalent of “I’d like to hire you to divide Excel into two completely separate and independent programs each of which has half (meaning roughly two-thirds) of the functionality of the original, based on these two top-level functions”, and you’re getting the idea.
Putting them into a body afterwards, of course, is a rounding error by comparison.