If you complete the tasks of the baby and deliver it back to her mum he gives you one of the 'baby demon dolls'
If you don't complete the tasks on time you can only take it as 'baby infant' never using it, as it. As soon as you use it he will dissapear disappear 'returning back to her mum'. So yes, it is posible possible to keep it as demon infant, just make sure you dont don't click it.