Most comments got it right, in the backstory Lilith is a piece of Morrigan's soul that was separated from her so her power wouldn't consume the demon world like a new born Galactus before she was ready for it. Canonically they fused back together and the fused form shows up in cross overs post D3.
Lilith is just essentially "leftover" power from Morrigan. She was never intended to be her own character (in lore), just essentially what happens when you pour Morrigan's power into a cup, and some flows out at the end. That's why she recombines, after Morrigan "gets a bigger cup", essentially.
In Darkstalkers 2 Lilith's ... is that Alek(?) umm... I think Jo might be on to something here. I think this deserves to be explored deeper. I mean further. Like as far as it can go.
I tought Lilith was some kind of clone... I remember one of their endings Lilith's form is vanishing and she ends up "possessing" Morrigan through a magic ring. The MvC series even references it by having Morrigan's duplicate during some moves be Lilith, instead of just a mirror copy.
...I mean...isn't the lore just that Lilith is just a power seal for Morrigan, and if they were to fuse, Morrigan would become the most powerful demon in hell or something?