You do have to use (in my experience at least) the merging chip, but both the AI and Aurora consent to this. Mangle and her AI come to an agreement and merge on their own without any outside help. Roxy's AI has to be removed as it is trying to take her over. Chica's AI consents to removal.
In summary, Roxy's AI is the only one who suffers.
The girls do still keep their personalities, as the AI and the soul have the same personality, and everything is consented to. From what I have understood, only Aurora merges with her AI, as Mangle and her AI agree to coinhabit the same body, and Chica and Roxy have their AI's removed completely