I once received a great and beautiful healing from a cat who was a Reiki Master. And he was an impeccable, fearless and inspired Master, indeed. Attuned by his owner, a good friend.
If you are asking the question, it could be the pet has already asked you intuitively for the attunement. I do suggest you do whatever you can to ascertain permission. And do your best to honor the attunement process as you learned it, while modifying for obvious differences.
I read of Reiki cats sometime this past year, email me for the publication reference if you want.
Thank you for reminding me of that venerable cat; it is worth meditation this evening.
Blessings,
Aronaya
P.S. I see treatments as very different from attunements. The level of permission and intent is very different. With a treatment, the being is consenting to receiving Reiki that one time. With an attunement, the being is signing up for a lifetime of having a changed energy field. I do use "attunement" and "initiation" to mean the same thing, but treatment is different.