Horcruces in canon don't seem to affect people's personalities or magic

w/r/t magic: IIRC, the Voldemort-Horcrux inside Harry is responsible for him being able to speak Parseltongue. (Although my memory is a bit fuzzy on this; it’s been a while since I last read book 7.) Also, there’s a weird horcrux/magic interaction, which leads to the King’s Cross scene near the end of book 7.

w/r/t personality: Wearing a Horcrux can at least influence the mood of the wearer, see the big row between Harry/Hermione and Ron in book 7. (Why canon!Harry is apparently not influenced by being a horcrux is still a mystery to me …)

Here, though, Harry isn't just a vessel for Voldemort; he's actually a copy of the same soul. It makes perfect sense for the wands to be very similar, and it makes sense from an authorial perspective to emphasize that: it looks like an echo of canon in the event, but it's the sort of thing that enriches the story on a second reading.

You’re absolutely right; the Law of Conservation of Detail is already satisfied by that. However, I still consider Priori Incantatem a plausible option for the next few chapters.