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Take the pet to a mirror, and you should be able to change quite a lot.
Edited to add: It's possible certain mods might let you change other things (kinda like the surgical station for sims?) but I'm not sure which mods. Also not sure if any of the changes done to a pet through a mirror or mod will be genetic.
Change appearance on a pet is like taking a sim to the mirror all you can change is their hair colour and style etc. All it does is let you change their fur colour and add spots etc and change their collars. So no you cant change a pets breed once you have cas or the game made the pet as a stray or adoptable.