Obsessive about one’s pet? (6)
I believe the answer is:
animal
'pet?' is the definition.
(pet is a kind of animal)
'obsessive about one's' is the wordplay.
'obsessive' becomes 'anal' (anal can mean obsessed with order).
'about' means one lot of letters goes inside another.
'one's' becomes 'I'm' ('one is' is a formal way of saying 'I am').
'anal' going around 'im' is 'ANIMAL'.
(Other definitions for animal that I've seen before include "Animate being" , "One sharing our kingdom" , "Neither vegetable nor mineral" , "Such as 13 ac" , "See 26" .)