Indeed. She also didn't pick the symbol originally; it was present when she first learned Reynardine had hijacked her doll. I suspect it has to do with Etheric something-or-other that changes based on its owner's affiliations, likely stemming from the same place the Realm of the Dead (another natural Etheric construct) was pulling its symbols. Alternately, it is indeed Annie's father who changed the symbol by messing with the bindings. What's more interesting is that this is another point not only in how Kat's symbol is not unique, but also who might have shared it with her. I still suspect Diego, myself, though again it's mostly just wild speculation; the crucible also looking like a sideways "D" and his work on Etheric sciences in the form of the arrow that bound Jeanne aren't really much evidence.
I suspect most of Poppa's problems with Rey don't actually stem from Annie secretly rebelling. I suspect he just doesn't understand Etheric sciences or how the bindings actually work. He can't exactly ask the Donlans who seem to be the Court experts at present, since given the way he's acted and the very few things we know he's done so far (bone spikes), I suspect strongly that they would be disinclined to help him very much with whatever he's planning. It may just be, if he altered the nature of the bindings, that he screwed up with that, but at a completely wild guess, I wouldn't be surprised to find that Rey's bindings aren't so much positive as they are negative in nature - rather than compelling him to do something, they block him from doing other things. I'm not sure how many times Annie's actually given him direct orders, but I recall her ordering him to not to speak (before the bridge) and ordering him not to revert to doll form (during the revelations around Surma's relationship with the two of them). Most of her interactions with Rey have actually been based on mutual respect, due to her status as Surma's child and also as the little girl who was willing to sit with him under the stars. Basically, Carver's giving Rey faulty orders based on his own assumptions which the bindings can't actually force Rey to do, and doesn't understand why it won't work.