Cat, cat or dog, or dog (7)
I believe the answer is:
whippet
'dog' is the definition.
(whippet is a kind of dog)
'cat cat or dog' is the wordplay.
'cat' becomes 'whip' (cat o' nine tails or cat is a type of whip).
'cat or dog' becomes 'pet' (I've seen this in another clue).
'whip'+'pet'='WHIPPET'
'or' is the link.
(Other definitions for whippet that I've seen before include "racing animal" , "Dog related to 9 down" , "Fast runner" , "Small slender breed of dog like a greyhound" , "Greyhound relative" .)