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What kind of pet does she keep?
Literal
She [topic-は] pet [as-に] what [object-を] is-keeping [explanatory-のです] [question-か]?
The に here marks role/category — 'what does she keep as a pet?' Compare nominal-に uses like お土産に買う ('buy as a souvenir') or 記念に取っておく ('keep as a memento'). 飼う ('to keep, raise, breed') is reserved for animals — humans use 育てる. のですか packages this as an explanatory question, hinting at curiosity about a pre-existing situation rather than asking abruptly out of nowhere.