Hi Totoro,
I think that some animals can reincarnate, but don't know the rules on that. Long, long ago, in my Theosophy days (and also going by other occult sources), the idea I enbibed was that the members of different species/types of animals (in their wild state) were, in general, merely extensions or parts of a feline, canine, or etc. group soul for that species/type rather than each having an individual soul. However, a particular animal could become individuated (in terms of its soul) at some point. The means for this step was generally fuzzy with one exception: Extended contact and association with humans as pets, helpers, companions, etc. generally resulted in such individuation. This mostly jives with what one hears in this area in terms of the return of "Fido" in a subsequent puppy or foundling. However, I am (once again) just speaking from poorly remembered readings 50+ years ago.
Cordially,
S&S
PS--Overall, I assume that it is the love and affection, going both ways, that can develop between the animal and its owner that is the individuating factor.