I love the "Abigail is the Wizard's granddaughter" theory because it does a lot of cool things but doesn't sacrifice Caroline's character into being just... a cheater. It's clear that Pierre and Caroline have some issues, but Pierre also has some high heart dialogue about trying to impress Caroline, and there's a few nuggets here and there that do imply they care about one another. Most of all, they both love Abigail, and especially for Caroline, she wants Abigail to have a very normal life.
To me, it makes a lot of sense that if you grew up in a magical world, you desperately would want that normalcy. It almost seems like Caroline is obsessed with it. She is incredibly worried about Abigail's fascination in exploration, magic, monsters, all that good stuff. I think she is, because Caroline herself lived that life, and she wants to give Abigail the life she never had-- a peaceful, magic-free one.
I can hear the argument of, well, why does the Wizard not know exactly who his daughter is? He only suspects that his daughter lives in Pelican Town. Well... There's a few more things to consider. Mid-late game spoilers: the witch's hut and its statues, namely the one that wipes memories.
The Wizard definitely has some issues with his past marriages too, there's no question about that. He is invested enough in their past relationship that he wants to ask you questions about her when you do some post-cc quests. I think it is completely possible that select memories of his were wiped, and I think it's also possible that the same applies to Caroline. Hell, maybe the whole town, if we wanted to go that far.
No one in town really knows about the Wizard much other than he's... a Wizard. Whenever he comes up in dialogue, it is usually in a wary way, a fear of the unknown type way, but they trust him enough to let him help out with some town festivals. The only one implied to know about him is Linus, who similarly has no dialogue about Wizard, but he does talk about nature and its magical influence. I love the idea that the Witch tampered with everyone's memories except for Linus's, who lives outside of town and has a stronger connection to nature anyway.
My theory about this brings the assumption that the Wizard may have caught on over time that there are holes in his memory, and he begins to patch things little-by-little as he watches over the town. For Caroline, while she may not remember Wizard, she would remember her upbringing as a very mystical one and have that yearning for a normal life. I'm fully convinced that Pierre's line about being unsure if Abigail is his is a total red herring and he's just having very temporary doubts.
Of course though, this all is a theory too and it's just my own thoughts! Sorry if they're a bit jambled but I hope it got through. That's what's so fun about the world in this game; you can come to all sorts of conclusions.
