We have no cases of this, so we don't know.
All cases of wizards with any form of mental disorder, or psychological condition are shown to either not have wands, or to have had their wands taken away. If a wizarding child had multiple personality disorder they would most likely be in St. Mungo's or kept watch in their own homes.
We already know that Wizards can be chosen by multiple wands so adding in multiple personalities doesn't really change this possibility, but it most likely limits the wizard or witch from receiving a wand at all.
Professor Lockhart is locked up as we see here - due to losing his memory
“Listen to him,” said the Healer, taking Lockhart’s arm and beaming fondly at him as though he were a precocious two-year-old. “He
was rather well known a few years ago; we very much hope that this
liking for giving autographs is a sign that his memory might be coming back a little bit. Will you step this way? He’s in a closed ward, you
know, he must have slipped out while I was bringing in the Christmas
presents, the door’s usually kept locked . . . not that he’s dangerous!
But,” she lowered her voice to a whisper, “bit of a danger to himself,
Neville Longbottom's parents were tortured to insanity - they share the closed ward with Lockhart and others.
“My son and his wife,” she said, turning haughtily to Harry, Ron, Hermione, and Ginny, “were tortured into insanity by You-Know-Who’s followers.”
Finally, Dumbledore's sister Ariana, who appears to have some form of PTSD (or similar) disorder from being tormented by other children.
They forced their way through the hedge,
and when she couldn’t show them the trick, they got a bit carried
away trying to stop the little freak doing it.
‘It destroyed her, what they did: she was never right again. She
wouldn’t use magic, but she couldn’t get rid of it: it turned
inwards and drove her mad, it exploded out of her when she
couldn’t control it, and at times she was strange and dangerous.
But mostly she was sweet, and scared, and harmless.'