fur baby
n.— «The comparative review of recent history to the present, and the changing status of dogs from living outside the home as working dogs on farms and as guard dogs, or serving as a distraction for the children in the family and perhaps as a means to teach responsibility, to the far more elevated status of a true member of the family, with new descriptive language like “pet parents” and the more recent coinage “fur baby,” is interesting and instructive.» —“A dog-centric America? A book review” by Mary Haight in Chicago Examiner Apr. 12, 2009. (source: Double-Tongued Dictionary)
Further reading
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I have long referred to my cat as my furbaby (fur baby). Your definition is exclusive to pet dogs. I think it should be broadened to include pet cats, as well.