Obviously a cat is not exactly doggy (5)
I believe the answer is:
moggy
'not exactly doggy' is the definition.
I know nothing about this answer so I can't judge whether this works.
'obviously a cat' is the wordplay.
I cannot quite understand how this works, but
'cat' could be 'mog' (informal British term) and 'mog' is present in the answer.
This may be the basis of the clue (or it may be nonsense).
'is' is the link.
Can you help me to learn more?
(Other definitions for moggy that I've seen before include "Cat (sl.)" , "Puss" , "Pussy" .)