Not necessarily. Anything that doesn't see play at the table is irrelevant, and anything that does just demands come consistency. So if the NPC was statted out as a Berserker, the player needs create a PC with a great axe and reckless attack, but has freedom regards to other stuff (staying in the theme of the character). I don't know too many players that would be happy being assigned a PC outside of a con game.
First, I was assuming the player wanted to take on the ex-NPC, either by choice or because it'd be a long time before any replacement PC could otherwise be introduced. "Sorry, Bob, the party don't have revival in the field and it'll be a while before anything new can come in. If you wanna play one of the enemy survivors that the party's taking in* there's an Elf MU, a Part-Orc sneak, a Dwarf Fighter who seems a bit on the berserk side, and a Human Fighter - take yer pick."
* - assume this decision has already been made, for these purposes.
And if the NPC was statted out as a Berserker and described as being x-feet tall, male, heavily bearded, wearing leather armour and carrying a selecton of axes and short blades; and bellowed insults at the PCs in what's probably Norse then all of those things need to be in the PC version.
Far simpler to have a rough-notes character sheet you can just hand to the player, isn't it?
(I see this far more often these days when the party end up taking in defeated-but-decent survivors - or opponents who switched sides partway through - as NPC party members: it's a common occurrence, and if I've already got these guys done up as PCs anyway I don't then have to re-do them on the spot.)
(and yes, party NPCs are in all ways the same as PCs, and the players run them sometimes when their PC is out of action)